UC-NRLF 


512 


A  MANUAL  OF 
FIELD  ASTRONOMY 


BY 

ANDREW  H.  HOLT 

Instructor  in  Civil  Engineering  in  the  College  of  Applied  Science 
of  the  State  University  of  Iowa 


FIRST   EDITION 
FIRST  THOUSAND 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:   CHAPMAN  &  HALL,  LIMITED 

1917 


\\V 


Copyright,  1916,  by 
ANDREW  H.  HOLT 


PUBLISHERS  PRINTING  COMPANY 
207-217  West  Twenty-fifth  Street,  New  York 


PREFACE 

IF  the  reason  be  demanded  for  the  appearance  of  another  book 
on  Field  Astronomy  when  there  are  already  published  several 
excellent  works  on  the  subject,  it  may  be  stated  as  follows: 
That  although  any  one  of  them  may  serve  very  well  as  a  text 
for  a  comparatively  extended  study,  the  author  has  been  unable 
to  find  one  sufficiently  concise  to  fit  the  short  time  usually  al- 
lowed for  the  work  in  a  civil  engineering  course  which  would  still 
provide  enough  of  the  fundamentals  of  the  subject  to  enable 
the  reader  to  make,  intelligently,  the  observations  and  accom- 
panying computations  required  in  the  practice  of  general  engineer- 
ing and  surveying.  Something  is  needed  more  complete  than 
the  usual  'chapter  in  books  on  surveying  and  less  extensive  than 
most  texts  on  field  astronomy.  This  need,  which  is  acknowledged 
by  other  teachers  to  exist,  it  is  hoped  to  fill;  and  at  the  same 
time  it  has  been  attempted  to  provide  a  book  which  will  be  of 
service  to  engineers  and  surveyors  whose  practice  requires  that 
they  occasionally  make  astronomical  observations. 

To  this  end  the  discussion  of  fundamentals  has  been  made 
brief,  but  it  is  thought  sufficiently  thorough  for  the  purpose. 
Special  attention  has  been  given  to  the  matter  of  measurement 
of  time,  because  it  is  believed  that  this  causes  more  difficulty 
for  students  in  general  than  any  other  part  of  the  subject. 

In  the  selection  of  the  methods  described  for  the  determina- 
tion of  latitude,  azimuth,  time,  and  longitude,  care  has  been 
taken  to  choose  those  which  are  believed  to  be  most  capable 
of  producing  results  when  used  with  field  instruments  under 
ordinary  field  conditions.  Realizing  that  the  determination 
of  azimuth  is  more  frequently  required  than  any  other  obser- 
vation, more  methods  have  been  given  for  this  than  for  the 
other  problems. 

Each  observation  has  been  presented  essentially  as  follows: 
The  work  of  which  the  observation  consists  is  first  stated  briefly, 
followed  by  the  relations  and  theory  on  which  it  depends,  ac- 
companied by  such  explanation  as  seems  necessary.  The  pro- 
cedure is  then  outlined,  step  by  step,  under  the  general  headings: 
o  rr  rr  ,«io  A 


IV  PREFACE 

"  Computations  Preceding  Field  Work,"  "  Field  Work/'  and 
"  Computations  Following  Field  Work."  This  outline  is  supple- 
mented by  a  copy  (near  the  back  of  the  book)  of  the  field-notes 
and  computations  of  a  similar  observation. 

It  is  hoped  that  this  method  of  presentation  will  commend 
itself  not  only  to  the  student  but  to  the  engineer  in  practice. 

The  "  Summary  of  Observations"  in-  Chapter  XI  should  be 
useful  in  selecting  an  observation  or  in  determining  whether 
sufficient  data  are  at  hand  to  permit  an  observation  which  is 
under  consideration. 

In  Appendix  A  are  given  the  derivations  of  the  formulas  of 
Spherical  Trigonometry  which  are  needed  in  the  work,  and  in 
Appendix  B  is  a  brief  discussion  of  the  theory  and  use  of  the 
"  Solar  Attachment  "  for  the  engineer's  transit. 

No  excuse  is  made  for  the  omission  of  refinements  of  either 
theory  or  practice  which  are  not  required  in  work  done  with 
an  engineer's  transit  or  a  sextant. 

While  preparing  the  manuscript  the  author  has  studied  several 
of  the  existing  works  on  field  astronomy,  and  this  book  has 
profited  thereby,  acknowledgment  being  made  in  the  body  of 
the  book  whenever  anything  has  been  copied.  No  claim  is  made 
to  having  produced  anything  new;  but  merely  to  having  put 
well-known  facts  in  a  new,  and  it  is  hoped  useful,  form. 

The  thanks  of  the  author  are  due  to  Messrs.  W.  and  L.  E. 
Gurley  and  the  Bausch  &  Lomb  Optical  Company,  who  have 
furnished  cuts  for  the  book ;  to  the  Superintendent  of  the  U.  S. 
Coast  and  Geodetic  Survey,  who  has  permitted  the  use  of  tables 
from  Government  publications,  to  friends  who  have  given  advice 
and  suggestions,  and  among  these  particularly  to  Mr.  R.  B. 
Kittredge,  Assistant  Professor  of  Railroad  Engineering  in  the 
College  of  Applied  Science  of  the  State  University  of  Iowa,  who 
has  read  the  entire  manuscript,  very  much  to  its  improvement. 

A.  H.  HOLT. 
IOWA  CITY,  IOWA,  November,  1916. 


NOTATION 

0  =  Latitude. 
X  =  Longitude. 

Zn  =  Azimuth,  referred  to  true  north. 
Zs  —  Azimuth,  referred  to  true  south. 

t  —  Hour  angle. 
RA  =  Right  ascension. 
h  =  Altitude. 
5  =  Declination. 
Z  =  Interior  angle  of  the  astronomical  triangle  at  the 

zenith. 
P  =  Interior  angle  of  the  astronomical  triangle  at  the 

pole. 
S  —  Interior  angle  of  the  astronomical  triangle  at  the 

star. 

z  =  Co-declination  or  polar  distance,  90°  —  5. 
p  =  Co-altitude  or  zenith  distance,  90°  —  h. 
s  =  Co-latitude,  90°  -  <j>. 

k  =  lA(s+p+z)  =  y2  [270°  -  (</>  +  h  +  «)]. 
Sid.  T  =  Sidereal  time. 
Std.  T  =  Standard  time. 
LMT  =  Local  mean  time. 
LA  T  =  Local  apparent  time. 

A  =  Right  ascension  of  the  mean  sun. 
An  —  Right  ascension  of  the  mean  sun  at  local  mean  noon. 
E  =  Equation  of  time. 


CONTENTS 


PAGE 

PREFACE iii 

NOTATION v 

CHAPTER  I 

INTRODUCTORY 

1.  Field  Astronomy 1 

2.  The  Celestial  Sphere 1 

3.  Apparent  Motion  of  the  Heavenly  Bodies    ....  2 

4.  Definitions 2 

CHAPTER  II 

SYSTEMS   OF   CO-ORDINATES   AND   THE   ASTRONOMICAL  TRIANGLE 

5.  Spherical  Co-ordinates 5 

6.  The  Horizon  System— System  I 6 

7.  The  Equator  Systems — Systems  II  and  III       ...  6 

8.  Uses  of  the  Three  Systems 8 

9.  Relation  Between  the  Systems 9 

10.  Relation  Between  Systems  I  and  II 12 

11.  Relation  Between  Systems  II  and  III 12 

12.  Some  Common  Solutions  of  the  Astronomical  Triangle  13 

CHAPTER  III 

MEASUREMENT   OF  TIME 

13.  The  Unit  of  Measurement 16 

14.  Apparent  Solar  Time 16 

15.  Mean  Solar  Time 17 

16.  Relation  Between  Apparent  and  Mean  Solar  Time — 

The  Equation  of  Time 17 

17.  Astronomical  and  Civil  Time 19 

18.  Standard  Time 20 

19.  Sidereal  Time 21 

20.  Relation  Between  Sidereal  and  Mean  Solar  Intervals  of 

Time 21 

21.  Relation  Between  Sidereal  and  Mean  Solar  Time  at  a 

Given  Instant 23 


Vlll  CONTENTS 

CHAPTER  IV 

THE    AMERICAN   EPHEMERIS    AND    NAUTICAL   ALMANAC 

PAGE 

22.  The  Ephemeris  .      .      .-.-..•.,...      .  25 

23.  Interpolation      . .     .     .      .      .  27 

CHAPTER  V 

PROBLEMS  IN  CONVERSION  OF  TIME 

24.  To  Change  Local  Mean  to  Local  Apparent  Time    .      .  29 

25.  To  Change  Local  Apparent  to  Local  Mean  Time    .      .  30 

26.  To  Change  Standard  to  Local  Mean  Time  .      .    -,      .  31 

27.  To  Change  Local  Mean  to  Standard  Time  ....  31 

28.  To  Change  Standard  to  Local  Apparent  Time  ...  32 

29.  To  Change  Local  Apparent  to  Standard  Time  ...  32 

30.  To  Change  Local  Mean  Solar  to  Sidereal  Time       .      .  33 

31.  To  Change  Sidereal  to  Local  Mean  Solar  Time       .      .  34 

CHAPTER  VI 

OBSERVATIONS CORRECTIONS    TO    OBSERVATIONS 

32.  Objects  Observed — Methods  of  Naming  Stars  .      .   » ,  36 

33.  Circumpolar  Constellations      ........  36 

34.  Parallax  .      .    '.      .      .      .      .      .      .      .      .    :.      .      .  38 

35.  Refraction '/l  ".     .      .      .      .      .  40 

36.  Semi-diameter .      .  41 

37.  Instrumental  Errors      . 41 

38.  Sequence  of  Corrections .  43 

39.  Suggestions  for  Observing        .      .      .      ...     .     .  43 

CHAPTER  VII 

OBSERVATIONS    FOR   LATITUDE 

40.  Latitude  by  a  Circumpolar  Star  at  Culmination      .      .  48 

41.  Latitude  by  Meridian  Altitude  of  a  Southern  Star     .  51 

42.  Latitude  by  Meridian  Altitude  of  the,  Sun    ....  53 


CONTENTS  IX 

CHAPTER  VIII 

OBSERVATIONS   FOR   AZIMUTH 

PAGE 

43.  Azimuth  by  a  Circumpolar  Star  at  Elongation        .      .  56 

44.  Azimuth  by  Polaris  near  Elongation        .      .      ...  58 

45.  Azimuth  by  a  Circumpolar  Star  at  Any  Hour  Angle    .  60 

46.  Azimuth  by  an  Altitude  of  the  Sun  or  of  a  Star      .      .  62 

47.  Azimuth  by  Equal  Altitudes  of  a  Star 65 

CHAPTER  IX 

OBSERVATIONS    FOR   TIME 

48.  Time  by  Transit  of  a  Star 68 

49.  Time  by  Transit  of  the  Sun 70 

CHAPTER  X 

OBSERVATIONS    FOR    LONGITUDE 

50.  Longitude  by  Transportation  of  Timepiece  ....       72 

CHAPTER  XI 

SUMMARY    OF    OBSERVATIONS 

51.  Observations  for  Latitude 74 

52.  Observations  for  Azimuth 74 

53.  Observations  for  Time 76 

54.  Observations  for  Longitude 76 

APPENDIX  A 

SPHERICAL   TRIGONOMETRY 

Derivation  of  Formulas  Required  in  Field  Astronomy  .      .       77 
APPENDIX  B 

SOLAR   ATTACHMENTS    FOR   TRANSITS 

The  Solar  Attachment .  84 


X  CONTENTS 

TABLES 

PAGE 

I.  Conversion  of  Sidereal  into  Mean  Solar  Time     .      .  93 

II.  Conversion  of  Mean  Solar  into  Sidereal  Time      .      .  94 

III.  Mean  Refraction 95 

IV.  Sun's  Parallax  and  Semi-diameter 96 

V.  Local  Mean  (Astronomical)  Time  of  the  Culminations 

and  Elongations  of  Polaris  in  the  Year  1915,  with 
Corrections  for  Referring  the  Tabular  Quantities  to 
Other  Years 97 

VI.  Mean  Declinations  of  Polaris 100 

VII.  Azimuth  of  Polaris  when  at  Elongation  for  Any  Year 

Between  1915  and  1928 101 

VIII.  Corrections  for  Obtaining  Azimuth  of  Polaris  when 

Near  Elongation  from  Azimuth  at  Elongation       .     104 
Greek  Alphabet        105 

EXAMPLES  OF  FIELD-NOTES  AND  COMPUTATIONS    .     .     .     106 
INDEX  127 


A   MANUAL   OF   FIELD 
ASTRONOMY 

CHAPTER  I 
INTRODUCTORY 

1.  Field  Astronomy.     Practical  Field  Astronomy  for  the  engi- 
neer consists  of  the  theory  and  practice  of  the  determination  by 
observations  on  the  sun  and  the  stars  of:   (1)  Latitude,  (2)  Longi- 
tude,  (3)  Azimuth,  and  (4)  Time.     Occasionally  observations 
are  made  on  the  moon,  but  those  on  the  sun  and  the  stars  are 
the  most  important. 

The  engineer  is  not  concerned  with  much  that  goes  to  make 
up  the  science  of  Astronomy.  He  makes  measurements  of  the 
directions  of  the  heavenly  bodies;  but  takes  no  account  of  their 
distances,  their  actual  movements  in  space  or  their  physical 
characteristics.  They  are  to  him  simply  objects  of  known  posi- 
tions from  which  he  can  make  measurements  to  determine  his 
position  on  the  earth's  surface  or  to  orient  the  courses  of  his 
survey. 

2.  The  Celestial  Sphere.     Since  only  the  directions  of  the 
sun  and  stars  are  to  be  considered,  it  is  convenient  to  regard 
them  all  as  being  situated  on  the  surface  of  a  sphere  of  infinite 
radius,  called  the  celestial  sphere,  with  its  center  at  the  center 
of  the  earth.     For  most  of  the  work  the  earth  (having  a  finite 
radius  as  compared  with  the  infinite  radius    of   the   celestial 
sphere)  is  considered  to  be  a  point.     That  there  is  no  appre- 
ciable error  in  this  is  apparent  from  the  fact  that  the  ratio  of 
the  radius  of  the  earth  to  the  distance  to  the  nearest  fixed  star  is 
about  1  to  7,000,000,000. 

It  should  be  noted  that  since  the  radius  of  the  celestial  sphere 
is  assumed  to  be  infinite,  all  parallel  planes  which  are  separated 
by  any  finite  distance  may  without  appreciable  error  be  con- 
sidered to  cut  its  surface  in  the  same  circle;  and  all  parallel 
lines  may  be  assumed  to  pierce  it  in  the  same  point.  Also, 

1 


Z  INTRODUCTORY 

any  plane  through  the  earth  will  cut  the  surface  of  the  celestial 
sphere  in  H  great  circ've.  (Since  it  is  assumed  to  pass  through 
the  center  of  the  sphere.) 

3.  Apparent  Motion  of  the  Heavenly  Bodies.     Due  to  the 
daily  rotation  of  the  earth  about  its  axis,  all  of  the 'stars  and 
the  sun  appear  to  be  traveling  from  the  east  toward  the  west 
along  circles  on  the  surface  of  the  celestial  sphere,  making  one 
revolution    a    day.     The    earth's   axis,   produced,   would   pass 
through  the  centers  of  these  circles  and  would  be  perpendicular 
to  their  planes.     Due  to  the  earth's  eastward  motion  in  its 
orbit  around  the  sun,  once  a  year,  we  see  the  sun  at  different 
times  from  different  places  in  the  orbit,  and  therefore  in  ap- 
parently different  positions.     This  apparent  motion  of  the  sun 
is  eastward  along  a  great  circle  on  the  celestial  sphere  whose 
plane  passes  through  the  earth  and  makes  an  angle  of  about 
23°  27'  with  the  plane  of  the  equator;    and  it  amounts  to  one 
revolution  around  the  earth  in  one  year. 

In  astronomy  the  terms  "east"  and  "west"  are  often  used 
to  indicate  directions  of  rotation  instead  of  in  the  sense  with 
which  we  are  familiar  in  plane  surveying.  The  reason  for  this 
will  be  apparent  if  one  considers  two  persons  standing  on  opposite 
sides  of  the  earth  and  both  pointing  east  (or  west).  They  would 
actually  be  pointing  in  opposite  directions,  but  they  would  be 
indicating  the  same  direction  of  rotation. 

The  student  should  become  accustomed  to  thinking  of  relative 
positions  and  motions  of  objects  on  the  celestial  sphere  from 
both  an  inside  and  an  outside  point  of  view.  It  is  usually  easier 
to  visualize  these  things  if  one  imagines  himself  outside,  with 
a  general  view  of  the  whole  situation;  although,  of  course,  they 
will  actually  have  to  be  viewed  from  an  inside  position. 

If  one  imagines  himself  outside  the  celestial  sphere  and  di- 
rectly above  the  north  pole,  on  line  with  the  axis  of  rotation  of 
the  earth,  an  eastward  rotation  of  a  celestial  object  will  appear 
to  be  counter-clockwise,  while  a  westward  rotation  will  appear 
to  be  clockwise. 

In  general,  we  shall  study  apparent  and  not  real  motions; 
and  therefore,  whether  considering  one's  self  at  the  center  of  the 
sphere  or  outside  and  above  the  north  pole,  the  earth  is  usually 
assumed  to  be  standing  still  and  the  other  bodies  to  be  moving 
around  it. 

4.  Definitions.     The  following  are  some  of  the  terms  used  in 


INTRODUCTORY  3 

astronomy  in  connection  with  defining  the  positions  of  celestial 
objects.     The  letters  are  references  to  Fig.  1. 

The  direction  of  the  plumb-line  at  any  place  is  called  the 
vertical  for  that  place.  If  the  direction  of  the  vertical  be  pro- 
duced indefinitely,  both  up  and  down,  it  will  intersect  the  surface 
of  the  celestial  sphere  in  two  points,  called  the  zenith  and  the 
nadir,  respectively  (Z  andZ'). 


— /-/-*• -A-N 


South 


W 


'     >*'^--~. 


Z' 
FIG.  1.    THE  CELESTIAL  SPHERE. 

A  plane  through  the  earth  perpendicular  to  this  direction  will 
cut  the  surface  of  the  celestial  sphere  in  a  great  circle,  called 
the  horizon  for  that  place  (HWH'). 

If  the  axis  of  rotation  of  the  earth  be  produced  indefinitely 
it  will  pierce  the  surface  of  the  celestial  sphere  in  the  north  and 
the  south  celestial  poles  (P  and  P').  A  plane  perpendicular  to 
this  axis  at  the  center  of  the  earth  will  cut  the  surface  of  the 
earth  in  the  terrestial  equator  and  the  surface  of  the  celestial 
sphere  in  the  celestial  equator  (EVE' A). 


£b  INTRODUCTORY 

Vertical  circles  (ZSMZ')  are  great  circles  on  the  surface  of 
the  celestial  sphere,  passing  through  the  zenith  and  the 
nadir. 

Hour  circles  (PSRP')  are  great  circles  through  the  celestial 
poles. 

The  meridian  of  the  observer  (PZP'Z')  is  a  great  circle  through 
the  zenith  and  the  celestial  poles.  It  is  at  the  same  time  a 
vertical  circle  and  an  hour  circle.  The  projection  of  this  meridian 
upon  the  earth  is  the  meridian  used  in  plane  surveying. 

The  prime  vertical  is  the  vertical  circle  whose  plane  is  perpen- 
dicular to  the  plane  of  the  meridian.  It  cuts  the  horizon  in  the 
east  and  west  points.  The  meridian  cuts  the  horizon  at  the 
north  and  the  south. 

The  great  circle  on  the  celestial  sphere  which  the  sun's  center 
appears  to  describe  in  its  (apparent)  yearly  motion  around  the 
earth  is  called  the  ecliptic  (Q'VQA),  and  the  angle  which  its  plane 
makes  with  the  plane  of  the  equator  (about  23°  27')  is  called  the 
obliquity  of  the  ecliptic.  The  points  at  which  the  ecliptic  inter- 
sects the  equator  are  called  the  equinoxes.  The  one  at  which  the 
sun  appears  to  cross  the  equator,  going  northward,  about  March 
21  of  each  year,  is  called  the  vernal  equinox  (F);  and  the  one  at 
which  it  crosses,  going  southward,  about  September  22,  is  called 
the  autumnal  equinox  (A). 

The  latitude  of  a  place  may  be  defined  as  the  angular  distance 
of  the  place  north  or  south  of  the*  equator;  or  more  exactly,  as  the 
angle  which  the  vertical  at  the  place  makes  with  the  plane  of  the 
equator.  Its  limiting  values  are  zero  and  plus  or  minus  ninety 
degrees.  North  latitudes  are  considered  plus  and  south  latitudes 
minus. 

The  longitude  of  a  place  is  the  angular  distance  (expressed  in 
either  degrees  or  hours)  of  the  place  east  or  west  of  some  arbi- 
trary reference  meridian,  usually  the  meridian  of  Greenwich, 
England.  More  exactly,  it  is  the  angle  between  the  planes  of  the 
reference  meridian  and  the  meridian  of  the  place.  Its  limits  are 
zero  and  180°  (or  12  hours)  east,  and  zero  and  180°  (or  12  hours) 
west. 

Fig.  1  illustrates  the  relative  positions  of  the  lines  and 
points  defined  above. 


CHAPTER   II 

SYSTEMS   OF   CO-ORDINATES  AND   THE 
ASTRONOMICAL  TRIANGLE 

5.  Spherical  Co-ordinates.  The  work  of  field  astronomy 
requires  that  we  shall  be  able  to  define  the  positions,  or  more 
exactly,  the  directions  of  the  heavenly  bodies.  For  this  purpose 
four  systems  of  spherical  co-ordinates  are  used.  These  systems 
have  several  characteristics  in  common.  They  are  all  systems 


FIG.  2. 


of  polar  co-ordinates,  with  the  earth  at  the  center  or  pole.  In 
each  system  the  direction  of  the  point  in  question  is  located  by 
means  of  two  angles,  or  arcs.  One  is  measured  along  a  primary 
circle  from  some  starting-point  to  the  foot  of  a  secondary  circle 
through  the  point  to  be  located.  The  other  is  measured  along 
the  secondary  circle  from  the  primary  circle  to  the  point  to  be 
located.  The  plane  of  the  secondary  circle  is  always  perpen- 
dicular to  the  plane  of  the  primary  circle. 

In  Fig.  2  the  direction  of  point  C  is  determined  by  the  angle 
AOB,  or  the  arc  AB  measured  from  A  along  the  primary  circle 
ABD  to  B,  and  the  angle  BOC,  or  the  arc  BC  measured  along 

5 


6 


CO-ORDINATES   AND    ASTRONOMICAL   TRIANGLE 


the  secondary  circle  from  the  primary  circle  to  point  C.  The 
plane  of  the  arc  BC  is  perpendicular  to  the  plane  of  the  arc  AB. 
Note  that  only  the  direction  of  the  point  is  determined;  no 
account  is  taken  of  its  distance — of  the  length  of  the  radius. 
This  method  of  locating  points  is  common  to  the  four  systems. 
Of  the  four  we  shall  use  three. 

6.  The   Horizon   System.     System    I,   sometimes   called   the 
Horizon  System,  has  for  its  primary  circle  the  horizon,  and  for 


its  secondary  circle  a  vertical  circle  through  the  point  to  be 
located.  The  primary  co-ordinate,  azimuth,  is  measured  from 
the  point  of  intersection  at  the  south  of  the  observer's  meridian, 
and  the  horizon,  westward  (clockwise)  along  the  horizon  to  the 
foot  of  a  vertical  circle  which  passes  through  the  point  to  be 
located.  The  secondary  co-ordinate  is  altitude,  and  is  measured 
along  the  vertical  circle  from  the  horizon  to  the  point. 

Fig.  3  shows  the  location  of  a  star,  S,  by  its  azimuth,  HM, 
and  its  altitude,  MS. 

In  some  special  cases  it  is  more  convenient  to  measure  the 
azimuth  from  the  north  instead  of  from  the  south,  and  it  is 
occasionally  so  measured. 

7.  The  Equator  Systems.  Systems  II  and  III  both  have  for 
their  primary  circle  the  celestial  equator,  and  for  their  secondary 
circle  an  hour  circle  through  the  point  to  be  located, 


THE   EQUATOR   SYSTEMS  7 

In  System  II  the  primary  co-ordinate,  hour  angle,  is  measured 
from  the  point  of  intersection  at  the  south  of  the  observer's 
meridian  and  the  celestial  equator,  westward  (clockwise)  to  the 
foot  of  the  hour  circle  through  the  point.  The  secondary  co- 


ordinate is  declination,  measured  along  the  hour  circle  from  the 
celestial  equator  to  the  point.  It  is  considered  plus  if  measured 
toward  the  north  celestial  pole  and  minus  if  toward  the  south. 

In  System  III  the  primary  co-ordinate  is  called  right  ascen- 
sion. It  is  measured  from  the  vernal  equinox  eastward  (counter- 
clockwise) along  the  celestial  equator  to  the  foot  of  the  hour 
circle  through  the  point  to  be  located.  The  secondary  co-ordi- 
nate is  the  same  as  that  of  System  II.  The  limiting  values  of 
hour  angle  and  of  right  ascension  are  in  each  case  0  hours  and 
24  hours.  The  limiting  values  of  declination  are  zero  and  plus 
or  minus  ninety  degrees. 

Fig.  4  illustrates  the  location  of  a  point  by  System  II,  and 
Fig.  5  shows  the  use  of  System  III. 

Some  attention  should  be  given  to  fix  in  mind  the  full  meaning 
of  the  term  "  hour  angle."  It  is  the  value  in  hours  (15°  per 
hour)  of  the  spherical  angle  EPS  (Fig.  4)  or  of  the  arc  ER. 
Remembering  that  the  point  S  is  traveling  westward  (clockwise) 
along  a  circle  whose  plane  is  parallel  to  the  plane  of  the  equator, 
it  will  be  seen  that  the  hour  angle  represents  the  number  of 
hours  since  the  point  crossed  the  meridian  of  the  observer. 


8        CO-ORDINATES   AND   ASTRONOMICAL   TRIANGLE 


For  the  fourth  system  of  spherical  co-ordinates  we  shall  have 
no  use  in  practical  field  astronomy. 


Systems  I,  II,  and  III  are  summarized  in  the  following  table, 
and  this  summary  should  be  thoroughly  memorized: 


co 

Primary 
Circle 

Secondary 
Circle 

PRIMARY  CO-ORDINATE 

SECONDARY  CO- 
ORDINATE 

Name 

Meas.  from 

Direction 

Limits 

Name 

Limits 

I 

Horizon 

Vertical 
Circle 

Azimuth 

South 

Westward 

0°  to  360° 

Altitude 

0°to90° 

II 

Celestial 
Equator 

Hour 
Circle 

Hour 
Angle 

Intersection 
at  the  South 
of  Meridian 
&  Celes.  Eq. 

Westward 

OHo24& 

Declination 

±90° 

III 

Celestial 
Equator 

Hour 
Circle 

Right 
Ascension 

Vernal 
Equinox. 

Eastward 

Ofeto24>> 

Declination 

±90° 

8.  Uses  of  the  Three  Systems.  Each  of  the  three  systems 
possesses  certain  characteristics  in  which  the  others  are  lacking 
which  give  it  a  place  in  the  work  o"f  field  astronomy. 

System  I  is  the  system  most  used  in  field  measurements  of 
the  positions  of  the  heavenly  bodies.  The  reason  is  that  its 
co-ordinates,  the  azimuth  and  altitude  of  the  point  in  question, 
may  both  be  measured  directly  with  the  engineer's  transit.  On 
the  other  hand,  because  of  the  rotation  of  the  earth,  the  azimuth 


RELATION    BETWEEN   THE    SYSTEMS  9 

and  altitude  of  a  given  point  are  continually  changing;  and 
these  co-ordinates  also  depend  on  the  position  of  the  observer. 
This  system  is  therefore  undesirable  for  permanent  records  of 
the  positions  of  the  heavenly  bodies. 

In  System  II  the  first  of  these  difficulties  is  done  away  with. 
The  celestial  equator,  from  which  the  secondary  co-ordinate 
(declination)  is  measured,  lies  in  a  plane  normal  to  the  axis  of 
rotation  of  the  earth;  and  is  therefore  independent  of  that 
rotation.  Since  the  equator  is  also  independent  of  the  observer's 
position,  the  declination  of  a  fixed  point,  such  as  a  fixed  star,  is 
independent  of  the  time  (i.e.,  of  the  rotation  of  the  earth)  and 
of  the  observer's  position.  It  is  very  nearly  a  constant  quantity. 
Any  variation  may  be  computed,  so  that  the  declination  of  a 
heavenly  body  at  a  given  time  may  be  regarded  as  a  permanent 
record.  The  primary  co-ordinate  of  System  II  (hour  angle) 
increases  uniformly  with  the  time,  and  may  therefore  be  meas- 
ured with  a  watch  or  chronometer. 

In  System  III  the  point  of  reference  from  which  the  primary 
co-ordinate  is  measured  shares  in  the  apparent  rotation  of  the 
celestial  sphere,  so  that  the  right  ascension  of  a  fixed  point  does 
not  change  with  time.  There  are  some  slight  changes  in  the 
right  ascensions  of  the  fixed  stars,  due  to  a  slight  movement  of 
the  vernal  equinox.  The  amounts  of  these  slight  variations 
may  be  computed;  so  that  the  right  ascension  of  a  fixed  point, 
once  determined,  may  always  be  regarded  as  a  known  quantity. 
Since  both  co-ordinates  of  this  system  are  independent  of  the 
time  and  of  the  observer's  position  and  are  nearly  constant 
(the  amount  of  any  variation  being  obtainable),  they  are  suit- 
able for  use  as  permanent  records  of  the  positions  of  the  heavenly 
bodies;  and  they  are  so  used  in  the  "American  Ephemeris  and 
Nautical  Almanac"  and  similar  works.  Here  are  tabulated  the 
right  ascensions  and  the  declinations  of  the  sun,  planets,  moon, 
and  several  hundred  of  the  fixed  stars. 

9.  Relation  between  the  Systems.  Since  all  three  systems 
of  co-ordinates  have  their  uses  in  the  work  of  field  astronomy, 
_i+  is  essential  that  we  be  able  to  translate  from  one  system  to 
another.  It  is  in  this  connection  that  the  assumption,  that  the 
heavenly  bodies  are  situated  on  the  surface  of  a  celestial  sphere 
comes  into  play;  for  if  arcs  of  great  circle  are  considered  drawn 
through  the  proper  points,  forming  a  spherical  triangle,  most 
of  the  problems  in  transformation  of  co-ordinates  become  simply 
problems  in  spherical  trigonometry.  This  spherical  triangle  is 


10       CO-ORDINATES   AND   ASTRONOMICAL   TRIANGLE 

always  formed  (see  Fig.  6)  by  an  arc  of  the  meridian  of  the 
observer  (PZ),  an  arc  of  a  vertical  circle  through  the  point 
to  be  located  (ZS),  and  an  arc  of  an  hour  circle  through  that 
point  (PS). 

This  triangle  is  so  much  used  in  the  work  of  field  astronomy 


that  it  is  called  the  astronomical  triangle;  or  sometimes,  from 
the  letters  at  its  vertices,  the  "SPZ"  triangle. 

It  is  essential  that  we  become  thoroughly  familiar  with  the 
quantities  that  go  to  make  up  the  parts  of  this  triangle.  In 
the  northern  hemisphere  the  vertices  are  always  at  the  north 
celestial  pole,  the  observer's  zenith,  and  the  star  or  other  point 
whose  co-ordinates  are  under  consideration. 

The  angle  P  at  the  pole  is  always  the  hour  angle  of  the  star 
S  if  the  star  is  on  the  western  side  of  the  meridian,  as  shown  in 
Fig.  6;  or  it  is  equal  to  24  hours  minus  the  hour  angle  if  the 
star  is  on  the  eastern  side  of  the  meridian,  as  in  Fig.  7. 

The  angle  Z  at  the  zenith  is  equal  to  (180°  —  Z3)  if  the  star 
is  on  the  western  side  of  the  meridian,  as  in  Fig.  6;  or  to  (Zs  — 
180°)  if  it  is  on  the  eastern  side  as  in  Fig.  7. 

The  angle  S  at  the  star  is  called  the  parallactic  angle.  It  is 
very  little  used. 

The  arc  ZE  is;  by  definition,  the  observer's  latitude;    and 


RELATION   BETWEEN   THE    SYSTEMS 


11 


therefore  the  arc  PZ,  or  s,  is  equal  to  (90°  —  0),  and  it  is  called 
the  co-latitude. 

The  arc  MS  is  the  altitude  of  the  star,  so  that  the  arc  SZ, 
or  p,  is  equal  to  (90°  —  h),  and  is  called  the  co-altitude,  or 
sometimes  the  zenith  distance. 

The  arc  RS  is  the  declination  of  the  star,  and  therefore  the 


FIG.  7. 

arc  PS,  or  z,  is  equal  to  (90°  —  5)  and  is  called  the  co-declination, 
or  sometimes  the  polar  distance. 

Thus  each  part  of  the  astronomical  triangle,  with  the  exception 
of  the  angle  at  the  star,  may  be  expressed  in  terms  of  the 
observer's  position  on  the  earth's  surface  (latitude)  or  the 
co-ordinates  of  the  star. 

It  may  be  convenient  to  note  for  use  in  future  solutions  of 
the  astronomical  triangle  for  hour  angle  or  for  azimuth,  that  if 
t  is  less  than  12  h,  or  180°,  Zs  is  less  than  180°;  and  if  t  is  greater 
than  12  h,  or  180°,  Zs  is  greater  than  180°. 

The  values  of  the  five  parts  of  the  astronomical  triangle  defined 
above  are  summarized  in  the  following  equations: 
s  =  90°  -  0    .      .      .      . 
p  =  90°  -  h    .      .      .      . 
z  =  90°  -  5     .      .      .      . 
Z  =  180°  -  Zs      ... 
or 
-  Zs  -  180°      . 


(14) 
(15) 
(16) 
(17) 

(17a) 


12       CO-ORDINATES   AND   ASTRONOMICAL   TRIANGLE 

P  =  t     ........       (18) 

or 
=  360°  -  t  ......     (18a) 

10.  Relation  between  Systems  I  and  II.  If  the  latitude  of  the 
place  is  known,  and  it  is  required  to  change  from  System  I 
to  System  II,  the  problem  becomes: 

Given,  in  the  astronomical  triangle:     s,  p,  Z. 

Required:     P,  z. 

Three  parts  of  the  spherical  triangle  being  given,  it  may  be 
solved  for  the  two  parts  required;  using  equation  (1)  (from 
Appendix  A),  to  find  the  side  z,  and  then  Equation  (3)  to  find 
the  angle  P. 

If  it  is  required  to  change  from  System  II  to  System  I,  the 
problem  is: 

Given,  in  the  astronomical  triangle:    s,  z,  P. 

Required:     p,  Z. 

Again  three  parts  of  a  spherical  triangle  are  known,  and  the 
triangle  may  be  solved  for  the  two  required,  obtaining  first  the 
side  p  and  then  the  angle  Z  by  the  two  equations  mentioned 
above. 

Solution  for  Zs  and  h  from  <f>,  t,  and  5  may  be  made  directly 
by  means  of  the  following  formulas  from  Chauvenet's  "  Spherical 
and  Practical  Astronomy,"  Vol.  I,  Article  14: 

cos  M  .  tan  t 

tan  Zs  =  -—  -  —      ....      (19) 
sin  (<£  —  M) 

tenh-  Z'  ....      (20) 

—  M) 


where  M  is  such  an  angle  that  : 

tan 


tan  M  = 


cos 


11.  Relation  between  Systems  H  and  III.  Since  in  the  second 
and  third  systems  the  secondary  co-ordinates  are  the  same 
(i.e.,  declination  in  each  case),  the  problem  of  changing  from 
one  system  to  the  other  becomes  merely  one  of  changing  hour 
angle  to  right  ascension,  or  vice  versa. 

Recalling  the  definitions  of  these  quantities  and  referring  to 
Fig.  8,  we  see  that  the  arc  ER  is  the  hour  angle  of  the  body  S, 
and  that  the  arc  VR  (V  being  the  vernal  equinox)  is  its  right 
ascension. 


RELATION   BETWEEN    SYSTEMS   II  AND   III 


13 


The  arc  EV  may  be  regarded  as  the  hour  angle  of  the  vernal 
equinox,  or  read  in  the  direction  VE,  as  the  right  ascension  of 
the  meridian.  It  is  evident  that  the  arc  ER  plus  the  arc  VR 
is  equal  to  the  arc  EV.  This  relation  always  holds  true,  no 


matter  what  the  position  of  the  point  S  may  be;  provided  that 
when  necessary  we  add  24  hours  (or  360°)  to  the  hour  angle  of 
the  vernal  equinox. 

It  is  a  very  important  principle  that: 

The  right  ascension  plus  the  hour  angle  of  any  body  is  equal 
to  the  hour  angle  of  the  vernal  equinox,  or  to  the  right  ascension 
of  the  meridian. 

We  shall  learn  a  little  later  that  the  hour  angle  of  the  vernal 
equinox  is  called  "Sidereal  Time";  and  having  studied  the 
measurement  of  time  it  will  be  apparent  how  this  important 
principle  comes  into  play,  not  only  in  the  transformation  of 
co-ordinates  but  in  a  large  share  of  the  problems  that  we 
shall  have  to  solve. 

If  changes  between  Systems  I  and  III  are  desired  they  may 
be  made  through  the  medium  of  System  II. 

12.  Some  Common  Solutions  of  the  Astronomical  Triangle. 
Two  problems  which  occur  so  frequently  in  the  work  as  to  de- 
serve special  mention,  and  which  call  for  solutions  of  the  as- 
tronomical triangle,  are: 


14-       CO-ORDINATES   AND   ASTRONOMICAL   TRIANGLE 

(1)  Knowing  the  latitude,  and  having  given  the  declination 
and  altitude  of  a  body,  to  find  its  hour  angle  and  azimuth. 

For  computing  the  angle  P  of  the  astronomical  triangle,  from 
which  the  hour  angle  may  be  obtained,  any  one  of  the  following 
formulas  (from  Appendix  A)  may  be  used: 


.  cos  (k  +</>).  cos  (k  +  5) 

sin  -r*  =  \   .      .       (5) 

cos  <f>  .  cos  5 


P  sin  k  .  cos  (k  +  h) 

cos  —  =  \  ....        (7) 

2         \        cos  <£  .  cos  5 


cos  (k  +  </>)•  cos  (k  H-  S) 
tan    "   ==  X'        sin  k  .  cos  (k  +  h) 

For  computing  the  angle  Z,  from  which  to  obtain  the  azimuth, 
Zs  or  Zn,  any  of  the  following  (from  Appendix  A)  may  be  used : 


,  cos  (k  +  <ft)  .  cos  (k  +  h) 

sin  -7-  =  \l .      .        (4) 

cos  0  .  cos  h 


Z_          I  sin  k  .  cos  (k  +  5) 
2        \       cos  <t>  .  cos  h 

rn.Q  fir  4-  h) 

•    .      .        (8) 


sin  k  .  cos  (k  +  6) 

When  selecting  a  formula  from  which  to  determine  an  angle 
it  should  be  remembered  that  if  the  angle  is  near  90°  the  value 
may  be  found  more  accurately  through  its  cosine,  while  for  a 
small  angle  the  sine  gives  the  greater  precision.  More  precise 
than  either,  because  of  the  rapid  variation  of  the  function,  are 
the  tangent  formulas. 

(2)  The  second  solution  of  the  astronomical  triangle  to  be 
noted  here  concerns  circumpolar  stars. 

If  the  co-declination  or  polar  distance  of  a  star  is  less  than 
the  latitude  of  the  observer  the  star  will  not  at  any  point  in 
its  daily  rotation  go  below  his  horizon;  but  would,  if  the  light 
of  the  sun  were  not  so  bright  as  to  obscure  it,  be  always  visible. 
Such  a  star  is  called  a  circumpolar  star. 

The  circle  which  any  circumpolar  star  appears  to  follow  in  its 
daily  motion  is  at  two  points  tangent  to  vertical  circles.  See 


SOLUTIONS   OF   THE   ASTRONOMICAL   TRIANGLE       15 

Fig.  9.  These  points  are  the  ones  at  which  the  star  appears 
farthest  east  (at  S  in  Fig.  9)  and  farthest  west  (at  Sf).  These 
two  positions  are  called  the  points  of  greatest  elongation. 


In  these  cases  the  astronomical  triangle  is  right-angled  at 
S  (or  S')j  and  the  formulas  from  which  the  azimuth  and  hour 
angle  for  this  position  of  the  star  may  be  obtained  are  (from 
Appendix  A)  : 


sin  Z  = 


cos  P  = 


COS  </> 

tan  </> 
tan  d 


(12) 


(13) 


CHAPTER  III 
MEASUREMENT   OF   TIME 

13.  The  Unit  of  Measurement.     The  unit  of  measurement  of 
time  is  based  upon  the  period  of  rotation  of  the  earth  about  its 
axis.     It  is  not  known  that  this  period  is  invariable;    but  the 
variations,  if  any  exist,  are  too  small  to  be  measured,  and  the 
rotation  is  considered  uniform.     The  differences  between  the 
several   methods   of   measuring  time   arise   from   the   different 
methods  of  counting  these  rotations.     This  much  is  common 
to  all:    that  the  counting  of  the  periods  of  rotation  is  done  by 
noting  successive  passages  of  some  reference  point  over  the 
meridian  of  an  observer. 

Every  point  on  the  celestial  sphere  crosses  any  given  meridian 
twice  during  each  period  of  rotation  of  the  earth.  The  instant 
when  any  point  is  on  the  same  half  of  the  meridian  as  the  zenith 
is  called  the  upper  transit  or  upper  culmination  of  the  point. 
The  instant  when  it  is  on  the  opposite  half  of  the  meridian  is 
called  the  lower  transit  or  lower  culmination  of  the  point .  Except 
in  the  case  of  the  circumpolar  stars,  which  never  go  below  the 
horizon,  the  upper  transit  is  the  only  one  visible;  and  unless 
otherwise  stated  it  is  the  one  that  is  meant  when  the  transit 
of  a  body  is  mentioned. 

14.  Apparent  Solar  Time.     The  most  common  methods  of 
measuring  time  are  based  on  the  use  of  the  sun  as  a  reference 
point  in  counting  the  rotations  of  the  earth.     The  interval  be- 
tween two  successive  upper  transits  of  the  sun's  center  over 
the  meridian  of  an  observer  is  called  an  apparent  solar  day,  and 
the  system  of  measurement  of  which  this  interval  is  the  unit 
is  called  apparent  solar  time.     The  instant  of  transit  at  any 
place  is  apparent  noon  for  that. place,  and  the  apparent  solar 
time  at  any  instant  is  the  hour  angle  of  the  sun's  center  at  that 
instant;  i.e.,  the  number  of   hours  since  the  sun's  center  crossed 
the  meridian.     It  is  the  time  as  given  by  a  sun-dial. 

Because  of  the  earth's  motion  around  the  sun  in  the  same 
direction  as  its  rotation  about  its  own  axis,  the  direction  of  the 
reference  point  is  continually  changing;  .and  the  interval  be- 
tween two  successive  transits  of  the  sun's  center  is  not  the  true 
period  of  rotation  of  the  earth.  Moreover,  since  the  rate  of 

16 


MEAN   SOLAR   TIME  17 

motion  of  the  earth  in  its  orbit  is  not  constant,  the  change  in 
direction  of  the  reference  point  is  not  constant;  and  therefore 
the  lengths  of  apparent  solar  days  are  different  at  different 
times  of  the  year. 

16.  Mean  Solar  Time.  To  avoid  the  inconvenience  of  a  unit 
of  variable  length,  use  is  made  of  the  convention  of  a  fictitious 
mean  sun.  This  fictitious  "sun"  is  assumed  to  have  a  motion 
around  the  earth,  the  summation  of  which  amounts  in  a  year 
to  exactly  the  same  as  the  apparent  motion  of  the  real  sun — 
to  one  complete  revolution.  The  essential  difference  is  that 
while  the  apparent  motion  of  the  real  sun  takes  place  along 
the  ecliptic  at  a  varying  rate,  the  assumed  motion  of  the  mean 
sun  takes  place  along  the  celestial  equator  at  a  constant  rate. 

A  mean  solar  day  is  the  interval  between  two  successive 
transits  of  the  mean  sun  over  the  same  meridian.  It  is  a  con- 
stant unit  and  is  equal  in  length  to  the  average  of  all  the  apparent 
solar  days  in  the  year. 

Mean  noon  at  any  place  is  the  instant  of  upper  transit  of  the 
mean  sun  at  that  place. 

The  mean  solar  time  at  any  place  is  the  hour  angle  of  the 
mean  sun  at  that  place  at  the  given  instant. 

16.  Relation  between  Apparent  and  Mean  Solar  Time — 
The  Equation  of  Time.  The  two  chief  causes  of  the  irregularity 
of  the  apparent  motion  of  the  sun,  which  in  turn  causes  the 
difference  between  apparent  and  mean  solar  time,  are  as  suggested 
above:  the  variable  rate  of  motion  of  the  earth  in  its  orbit 
around  the  sun,  and  the  inclination  of  the  plane  of  this  orbit 
to  the  plane  of  the  equator. 

The  earth's  orbit  is  elliptical  in  shape,  and  in  order  that  the 
earth  may  obey  the  laws  of  gravitation  in  its  motion — viz.,  that 
the  line  joining  it  at  any  point  in  its  path  to  the  sun  shall  sweep 
over  equal  areas  during  equal  intervals  of  time — it  is  necessary 
that  its  rates  of  motion  at  different  times  should  be  different. 
In  the  winter,  when  the  sun  is  nearest  the  earth,  the  rate  of 
angular  motion  of  the  "radius"  is  faster;  and  therefore  the 
apparent  solar  days  are  longer  at  that  time  of  the  year.  In 
the  summer,  for  a  similar  reason,  they  are  shorter.*  The  maxi- 

*  The  relative  length  of  apparent  solar  days  at  different  times  of  the1  year  is 
not  to  be  confused  with  the  relative  length  of  the  periods  of  daylight  at  these 
seasons.  The  inclination  of  the  earth's  axis  of  rotation  to  the  plane  of  its  orbit 
is  such  that  in  the  winter  when  the  earth  is  actually  nearest  the  sun  the  latter 
has  its  greatest  southern  declination;  i.e.,  it  is  farthest  south  of  the  equator, 
so  that  the  period  between  sunrise  and  sunset  in  the  northern  hemisphere  is 
shortest.  Since  the  rays  from  the  sun  strike  the  surface  of  the  northern  hemi- 
sphere so  obliquely  at  that  time,  we  have  our  coldest  weather. 


18 


MEASUREMENT    OF   TIME 


mum  difference  between  apparent  and  mean  time  due  to  this 
cause  alone  is  about  eight  minutes,  plus  or  minus.  Due  to  the 
second  cause  mentioned  there  may  be  a  maximum  difference  of 
about  ten  minutes. 

The  maximum  combined  effect  of  these  two  causes  is  a  little 
over  sixteen  minutes;  and  this  difference  between  apparent  and 
mean  time,  varying  in  amount  from  zero  to  the  maximum,  is 
called  the  equation  of  time.  It  is  continually  changing,  but 
its  value  for  any  given  instant  may  be  computed.  Data  for 
this  computation  are  found  in  the  " American  Ephemeris  and 
Nautical  Almanac,"  to  which  reference  has  already  been  made, 
and  which  will  be  described  more  fully  in  Chapter  IV. 

To  change  from  apparent  to  mean  time  or  vice  versa,  it  is 


only  necessary  to  add  or  to  subtract  the  equation  of  time  for 
the  instant  at  which  the  change  is  to  be  made,  addition  or  sub- 
traction being  determined  by  the  time  of  year  and  the  direction 
of  change. 

Reference  to  Fig.  10  may  help  to  make  clear  the  process  of 
changing  from  apparent  to  mean  solar  time,  or  vice  versa. 

Let  S  and  S'  represent  the  real  sun  and  the  fictitious  or  mean 
sun,  respectively.  Both  are  traveling  westward,  clockwise,  in 
their  daily  rotation — the  mean  sun  on  the  equator  and  the  real 


ASTRONOMICAL   AND   CIVIL  TIME  19 

sun  on  a  path  which  is  above  and  nearly  parallel  to  the  equator, 
and  which  might  be  likened  to  one  turn  of  a  spiral  or  to  one 
turn  of  a  helical  spring. 

Local  apparent  solar  time — the  hour  angle  of  the  real  sun, 
the  number  of  hours  since  the  real  sun  crossed  the  meridian — 
is  the  arc  ER.  Local  mean  solar  time — the  hour  angle  of  the 
mean  sun,  the  number  of  hours  since  the  mean  sun  crossed  the 
meridian — is  the  arc  ES'.  Either  the  mean  sun  or  the  real 
sun  might  be  in  advance,  depending  on  the  time  of  year.  The 
equation  of  time,  the  amount  by  which  one  is  in  advance  of 
the  other,  is  the  arc  RSf. 

Since  we  are  able  to  compute  the  value  of  the  equation  of 
time  for  any  desired  instant,  it  is  obvious  how  if  either  local 
apparent  time  or  local  mean  time  is  known  we  may  obtain 
the  other. 

The  determination  of  the  instant  of  time  in  one  system  of 
measurement  corresponding  to  a  given  instant  in  another  system 
is  always  a  matter  of  comparing  the  positions  of  the  reference 
points  in  the  two  systems.  Stated  in  other  words,  the  problem 
is:  Having  given  the  hour  angle  of  the  reference  point  of  one 
system,  it  is  required  to  find  the  hour  angle  of  the  reference 
point  of  the  other  system. 

Examples  of  changing  from  apparent  to  mean  time  and  vice 
versa  will  be  given  after  we  have  learned  more  about  the  "Ameri- 
can Ephemeris  and  Nautical  Almanac," — in  Chapter  V. 

17.  Astronomical  and  Civil  Time.  For  astronomical  purposes 
the  mean  solar  day  is  divided  into  twenty-four  hours,  beginning 
at  the  instant  of  mean  noon;  and  the  hours  are  subdivided  into 
minutes  and  seconds. 

For  ordinary  purposes  the  mean  solar  day  is  divided  into  two 
periods  of  twelve  hours  each:  P.M.  (post  meridiem),  beginning 
at  mean  noon  and  continuing  until  twelve  o'clock,  midnight; 
and  A.M.  (ante  meridiem),  from  midnight  until  mean  noon  of 
the  next  day.  The  civil  day  is  considered  to  extend  from  mid- 
night to  midnight.  The  astronomical  day  begins  at  mean  noon 
on  the  civil  day  of  the  same  date. 

Astronomical  time  as  well  as  civil  time  may  be  either  apparent 
or  mean. 

For  changing  from  one  scheme  of  division  of  the  solar  day 
to  the  other  the  following  rules  may  be  used: 

To  change  Astronomical  Time  to  Civil  Time: 

If  less  than  12  hours  call  it  P.M. 


20  MEASUREMENT   OF   TIME 

If  greater  than  12  hours  subtract  twelve  hours,  add  one  day 
to  the  date,  and  call  it  A.M. 

To  change  Civil  Time  to  Astronomical  Time: 

If  A.M.  add  12  hours,  drop  one  day  from  the  date  and  drop 
the  A.M. 

If  P.M.  drop  the  P.M. 

For  instance: 

July  6,  8  h.,  astronomical  time  =  July  6,  8  P.M.,  civil  time. 

May  11,  4  A.M.,  civil  time  =  May  10, 16  h.,  astronomical  time. 

18.  Standard  Time.  Since  local  mean  solar  time  at  any  in- 
stant is  the  hour  angle  of  the  mean  sun  at  that  instant,  it  is 
evident  that  all  places  not  on  the  same  meridian  will  have 
different  local  mean  times.  To  avoid  confusion  from  this  source, 
a  uniform  system  of  time  was  established  in  the  United  States 
in  1883.  The  country  was  divided  into  belts,  each  15°  or  one 
hour  of  longitude  wide,  each  belt  to  use  as  standard  time  the 
local  mean  time  of  a  central  meridian.  There  are  four  such 
belts  across  the  country,  using  the  times  of  the  75th,  90th,  105th, 
and  120th  meridians  (west  of  Greenwich).  Local  mean  time 
of  the  75th  meridian  is  called  Eastern  Time ;  of  the  90th,  Central 
Time;  of  the  105th,  Mountain  Time;  of  the  120th,  Pacific 
Time.  Some  of  the  eastern  provinces  of  Canada  use  the  local 
mean  time  of  the  60th  meridian,  called  Atlantic  Time. 

The  theoretical  boundaries  of  these  time  belts  have  been 
shifted  to  suit  local  convenience,  and  the  boundaries  now  depend 
largely  on  the  location  of  cities  and  railroads;  but  the  difference 
in  time  between  one  belt  and  the  next  is  always  one  hour. 

For  instance,  when  it  is  noon  at  Greenwich,  England,  it  is 
7  A.M.  by  Eastern  Time,  6  A.M.  by  Central  Time,  5  A.M.  by 
Mountain  Time,  and  4  A.M.  by  Pacific  Time. 

To  change  from  Standard  Time  to  Local  Mean  Time  at  any 
place : 

Express  the  difference  in  longitude  between  the  standard 
and  local  meridians  in  units  of  time,  and  if  the  place  is  east  of 
the  standard  meridian  add  this  difference  to  the  standard  time; 
if  west,  subtract. 

To  change  from  Local  Mean  Time  to  Standard  Time  at  any  place : 

The  procedure  is  exactly  the  reverse. 

To  determine  whether  to  add  or  to  subtract  a  "correction,"  it 
is  only  necessary  to  remember  that :  The  farther  east  a  place  is 
the  later  it  is  by  local  mean  time  at  any  instant.  Standard 
time  is  simply  local  mean  time  at  a  standard  meridian.  . 


SIDEREAL   TIME 


21 


Examples  of  the  conversion  of  local  mean  to  standard  time, 
and  vice  versa,  are  given  in  Chapter  V. 

19.  Sidereal   Time.     So   far   we   have   studied   methods   of 
measuring  time  which  are  based  on  the  use  of  the  sun  as  a  refer- 
ence point  for  counting  the  rotations  of  the  earth  about  its  axis 
— i.e.,  solar  time — and  these    are  the  methods  in  common  use 
for  most  purposes.     But  since  the  sun  appears  to  make  one 
revolution  a  year  around  the  earth,  causing  successive  transits 
of  the  sun's  center  over  the  same  meridian  to  occur  at  slightly 
different  points  in  the  earth's  rotation,  the  intervals  between 
successive  transits  are  not  a  true  measure  of  the  period  of  that 
rotation. 

For  astronomical  purposes  a  more  precise  determination  is 
needed,  and  sidereal  time  is  used.  The  sidereal  day  is  the  in- 
terval between  two  successive  upper  transits  of  the  vernal  equinox 
over  the  same  meridian,  and  the  sidereal  time  at  any  place  is 
the  hour  angle  of  the  vernal  equinox  at  that  place  at  the  given 
instant. 

If  the  vernal  equinox  were  a  fixed  point  the  sidereal  day  would 
be  an  exact  measure  of  the  period  of  the  earth's  rotation.  The 
equinox  has  a  slow  westward  movement,  but  it  is  so  slight  that 
the  length  of  a  sidereal  day  differs  from  the  true  period  of  one 
rotation  by  only  about  0*01;  and  as  sidereal  time  is  not  used 
over  long  intervals  (dates  are  always  kept  in  solar  time)  cumu- 
lative errors  are  avoided;  and 
the  sidereal  day  as  defined  above 
is  the  one  actually  used,  without 
correction. 

20.  Relation  between  Sidereal 
and    Mean    Solar    Intervals   of 
Time.     It  has  been  mentioned 
that  to  the  apparent  motion  of 
the  sun  around  the  earth  is  due 
the   difference  between  sidereal 
and    solar   time.     Reference    to 
Fig.  11  may  help  to  make  this 
relation  between  the  two  more 
clear.     Let  the  large  circle  rep- 
resent   the    celestial    equator,   along  which   the  mean   sun    is 
assumed  to  move  at  a  uniform  rate;    and  let  the  small  circle 
represent  the  earth.     Let  an  observer  be  at  O  on  the  earth 
when  the  sun  is  on  his  meridian  at  S.     Now,  while  the  earth  is 


FIG.  11. 


22  MEASUREMENT   OF   TIME 

making  one  revolution  in  the  direction  indicated  by  the  arrow, 
the  mean  sun  is  also  moving  in  the  same  direction;  so  that  its 
next  transit  over  the  observer's  meridian  takes  place  when  the 
observer's  position  has  revolved  through  nearly  361°  to  Of,  and 
the  mean  sun  is  at  S'.  At  the  next  transit  the  observer  is  at  0" 
and  the  mean  sun  at  S".  In  one  year  the  sun  has  apparently 
completed  one  revolution  around  the  earth  and  is  again  at  S, 
so  that  in  that  time  one  rotation  of  the  earth  has  not  been 
counted.  In  the  meantime  the  vernal  equinox,  having  remained 
practically  a  fixed  point,  has  registered  the  exact  number  of 
rotations  that  have  taken  place.  There  are,  therefore,  one 
less  solar  than  sidereal  days  in  a  year. 

The  length  of  the  "  tropical  year"*  has  been  determined  to  be 
365.2422  mean  solar  days,  and  since  there  are  one  more  sidereal 
than  solar  days  the  relation  between  the  two  is  given  by  the 
following  equations: 

365.2422  mean  solar  days  =  366.2422  sidereal  days    .      .      (21) 

1  mean  solar  day  =  1.0027379  sidereal  days       .      (22) 

1  sidereal  day  =  0.9972696  mean  solar  days  .      (23) 

Tables  I  and  II  are  arranged  for  conversion  of  sidereal  into 
mean  solar  intervals,  and  vice  versa;  and  are  more  convenient 
to  use  than  formulas. 

Tables  II  and  III  of  the  "American  Ephemeris  and  Nautical 
Almanac"  (near  the  end  of  the  book)  are  for  the  same  purpose. 

It  will  be  convenient  to  remember  that  a  mean  solar  hour 
is  about  ten  seconds  longer  than  a  sidereal  hour,  and  that  a 
mean  solar  day  is  about  3  min.  56  sec.  longer  than  a  sidereal 
day. 

It  must  be  kept  clearly  in  mind  that  the  difference  between 
the  units  of  the  two  systems — sidereal  and  solar — is  simply  this : 
A  sidereal  hour  as  a  unit  of  time  is  the  interval  of  time  re- 
quired for  the  vernal  equinox  to  pass  over  15°  or  one  "  hour"  of 
arc.  A  mean  solar  hour  as  a  unit  of  time  is  the  interval  of  time 
required  for  the  mean  sun  to  pass  over  15°  or  one  "  hour"  of  arc. 
The  difference  arises  solely  from  the  different  rates  of  apparent 
motion  of  the  two  reference  points.  It  is  a  difference  between 
two  intervals  of  time — not  between  two  unit  angles  or  arcs. 
That  is,  an  hour  angle,  for  instance,  is  not  measured  in  "sidereal 
hours"  or  "solar  hours  "  but  simply  in  "hours" — twenty-fourths 
of  a  circumference — or  in  degrees,  15°  per  hour. 

*  The  tropical  year  is  the  interval  of  time  between  two  successive  passages 
of  the  sun  through  the  vernal  equinox. 


SIDEREAL   AND   MEAN   SOLAR   TIME  23 

It  must  also  be  carefully  noted  that  the  relation  between 
solar  and  sidereal  intervals  of  time,  as  discussed  in  this  article, 
is  simply  a  comparison  of  units.  It  must  not  be  confused  with 
the  relation  between  mean  solar  and  sidereal  time  at  a  given 
instant,  discussed  in  the  next  article.  The  one  has  to  do  with 
the  relative  sizes  of  the  units  of  measurement,  the  other  concerns 
the  relative  positions  of  the  mean  sun  and  the  vernal  equinox 
at  a  given  instant. 

21.  Relation  between  Sidereal  and  Mean  Solar  Time  at  a 
Given  Instant.  We  have  learned  (Art.  11,  page  12)  that  the 
right  ascension  of  any  body  plus  its  hour  angle  is  equal  to  the 
hour  angle  of  the  vernal  equinox,  and  (Art.  19,  page  21)  that 
the  hour  angle  of  the  vernal  equinox  is  sidereal  time.  It  follows, 
then,  that: 

The  sidereal  time  at  any  instant  is  equal  to  the  right  ascension 
plus  the  hour  angle  of  any  body  at  that  instant,  or 

Sid.  T  =  R  A  +  t (24) 

This  is  probably  the  most  used  and  perhaps  the  most  im- 
portant of  all  the  equations  needed  in  field  astronomy. 

If  the  mean  sun  is  the  body  under  consideration  the  following 
equation  is  true: 

Sidereal  Time  =  Right  Ascension  of  Mean  Sun  -f  Hour  Angle 
of  Mean  Sun. 

But  the  hour  angle  of  the  mean  sun  is  local  mean  solar  time, 
therefore : 

Sid.  T  =  A  +  LMT (25) 

LMT  =  Sid.  T  -  A      .      .      .      .      (26) 

Granting  that  we  can  obtain  the  value  of  A  for  the  desired 
instant,  these  equations  will  enable  us  to  obtain  the  sidereal  time 
corresponding  to  any  given  instant  of  local  mean  solar  time,  or 
to  find  the  local  mean  solar  time  corresponding  to  any  given  in- 
stant of  sidereal  time. 

The  right  ascension  of  the  mean  sun  is  entirely  independent 
of  the  location  of  the  observer,  and  is  dependent  only  on  the 
absolute  instant  of  time.  At  some  instant  about  March  22  of 
each  year  the  mean  sun  is  at  the  vernal  equinox,  and  its  right 
ascension  is  zero.  Leaving  the  vernal  equinox,  it  moves  east- 
ward along  the  celestial  equator  at  a  constant  rate;  and  there- 
fore A  is  equal  to  zero  at  some  instant  about  March  22  of  each 
year  and  ijicreases,  constantly,  to  24  hours  (or  0  hours)  at  some 


24  MEASUREMENT   OF   TIME 

instant  on  March  22  of  the  next  year.  The  value  of  A  for  the 
instant  of  mean  noon  at  Greenwich  for  each  day  in  the  year  is 
given  in  the  " American  Ephemeris  and  Nautical  Almanac."  To 
find  the  value  of  A  for  any  instant,  it  is  necessary  to  find  the 
interval  of  time  that  has  elapsed  since  the  last  preceding  mean 
noon  at  Greenwich,  determine  the  increase  in  the  right  ascension 
of  the  mean  sun  during  that  interval,  and  add  that  increase  to 
the  value  taken  from  the  tables  for  the  instant  of  Greenwich 
mean  noon  preceding. 

This  constant  increase  in  the  right  ascension  of  the  mean 
sun  is  simply  the  gain  of  the  apparent  motion  of  the  vernal 
equinox  over  that  of  the  sun  during  the  interval  considered,  and 
it  is  therefore  equal  to  the  difference  between  sidereal  and  mean 
solar  time  for  that  interval.  In  other  words,  it  is  the  difference 
between  the  number  of  sidereal  units  in  the  interval  of  time 
and  the  number  of  mean  solar  units  in  the  same  interval.  This 
increase,  or  "correction,"  may  therefore  be  taken  directly  from 
Tables  I  and  II  at  the  back  of  this  book  or  from  Tables  II  and 
III  at  the  back  of  the  "  Nautical  Almanac." 

Problems  in  changing  from  sidereal  to  mean  solar  time  and 
from  mean  solar  to  sidereal  time  will  be  solved  in  Chapter  V. 

Changes  from  apparent  solar  time  to  sidereal  time  and  vice 
versa  may  be  made  by  first  changing  to  mean  solar  time  in  each 
case. 


CHAPTER  IV 

THE   AMERICAN   EPHEMERIS   AND 
NAUTICAL  ALMANAC 

22.  The  Ephemeris.  The  "American  Ephemeris  and  Nautical 
Almanac  "  is  published  yearly,  three  years  in  advance,  by  the 
Nautical  Almanac  Office  of  the  United  States  Naval  Observatory, 
at  Washington,  D.  C.,  and  is  sold  by  the  Superintendent  of 
Documents.  (See  note  on  page  28.)  It  contains  data  of  use  to 
surveyors,  navigators,  and  others  for  astronomical  calculations, 
made  up  from  the  results  of  observations  with  large  instruments 
at  the  principal  observatories,  and  from  calculations. 

These  data  comprise  ephemerides*  of  the  sun,  moon,  planets, 
and  stars,  the  equation  of  time,  semi-diameters,  and  horizontal 
parallaxes  of  heavenly  bodies,  convenient  tables  for  conversion 
of  units,  etc.,  as  well  as  a  great  deal  of  data  in  regard  to  eclipses 
and  other  phenomena  of  more  interest  to  astronomers  than  to 
surveyors.  Many  of  these  quantities  vary  with  the  time,  so 
their  values  are  given  for  regular  intervals  of  time  for  the  meridian 
of  Greenwich  or  of  Washington. 

The  Almanac  is  divided  into  three  parts.  Part  I  is  an  "  Ephem- 
eris for  the  Meridian  of  Greenwich."  The  data  in  Part  I  of 
most  interest  to  surveyors  is  the  ephemeris  of  the  sun.  There 
is  also  an  ephemeris  of  the  moon  which  may  be  of  occasional 
use.  In  the  ephemeris  of  the  sun  there  are  given  for  the  instant 
of  mean  noon  at  Greenwich  for  each  day  in  the  year:  The  sun's 
apparent  right  ascension  and  declination  with  the  hourly  variation 
in  each,  its  semi-diameter  and  horizontal  parallax,  the  equation 
of  time  with  its  hourly  variation  and  with  the  proper  algebraic 
sign  for  changing  from  mean  to  apparent  time  (the  opposite  sign 
would  be  used  in  changing  from  apparent  to  mean),  the  sidereal 
time,  or  right  ascension  of  the  mean  sun.  Note  that  this  last 
quantity  is  the  sidereal  time  at  Greenwich  mean  noon,  or  the 
right  ascension  of  the  mean  sun  at  Greenwich  mean  noon;  i.  e., 

the  instant  when  the  mean  sun  is  on  the  meridian  of  Greenwich. 
All  the  above-mentioned  data  are  tabulated  on  successive  left- 

*  By  "ephemeris"  is  meant  a  catalogue  of  the  positions  of  a  celestial  body 
at  equidistant  intervals  of  time,  usually  given  in  terms  of  the  body's  right 
ascension  and  declination. 

25 


26    AMERICAN   EPHEMERIS   AND   NAUTICAL   ALMANAC 

hand  pages,  beginning  on  page  2.  On  the  right-hand  pages  are 
given  other  data  for  the  corresponding  dates;  but  only  the  last 
column,  which  gives  the  mean  time  of  sidereal  noon  for  each 
day  (i.e.,  the  Greenwich  mean  time  when  the  vernal  equinox 
is  on  the  meridian  of  Greenwich),  is  likely  to  be  of  use  to  the 
surveyor. 

Of  the  data  in  Part  II,  "  Ephemeris  for  the  Meridian  of 
Washington/'  the  following  are  of  use  to  the  surveyor:  An 
ephemeris  of  the  sun  for  the  instant  of  Washington  apparent 
noon  for  each  day,  ephemerides  of  thirty-five  circumpolar  stars 
for  the  instant  of  upper  transit  at  Washington  on  each  day, 
and  ephemerides  of  825  other  stars  for  the  instant  of  upper 
transit  at  Washington  at  intervals  of  ten  days. 

The  ephemeris  of  the  sun  gives  the  right  ascension  and  declina- 
tion of  the  sun  at  the  instant  of  Washington  apparent  noon  with 
their  hourly  variations,  the  equation  of  time  with  its  hourly 
variation  and  with  the  proper  algebraic  sign  for  changing  from 
apparent  to  mean  time,  the  semi-diameter  of  the  sun,  the  sidereal 
time  required  for  the  semi-diameter  of  the  sun  to  pass  the  merid- 
ian, and  the  sidereal  time  of  mean  noon  at  Washington  for  each 
day  in  the  year. 

The  table  of  which  each  page  is  headed,  "Apparent  Places  of 
Stars,  19 — ,  Circumpolar  Stars,"  gives  for  the  time  of  upper 
transit  at  Washington  for  each  day  in  the  year  the  right  ascen- 
sion and  declination  of  thirty-five  circumpolar  stars.  The  table 
of  which  each  page  is  headed,  "Apparent  Places  of  Stars,  19 — ," 
gives  for  the  time  of  upper  transit  at  Washington  at  intervals  of 
ten  days  the  right  ascension  and  declination  of  each  of  825  other 
stars.  ^  • 

It  will  be  seen  upon  examination  of  the  tables  that  the  right 
ascensions  and  declinations  of  these  stars  change  so  slowly 
that,  though  they  have  been  computed  for  the  instants  of  upper 
transit  of  the  stars  at  Washington,  there  will  be  no  appreciable 
error  in  any  work  for  which  a  surveyor  is  likely  to  need  the 
data  if  these  values  are  used  for  the  upper  transit  on  the  cor- 
responding date  at  any  place  in  the  United  States.  The  dates 
in  the  column  at  the  left  of  each  page  of  the  star  ephemerides 
are  in  mean  solar  time;  and  each  contains  a  decimal,  as  April  3.7, 
July  17.1,  etc.  The  decimal  part  of  the  date  indicates  the 
approximate  time,  in  tenths  of  twenty-four  hours,  from  mean 
noon  of  the  day  indicated  by  the  integral  part  of  the  date  to 
the  time  of  upper  transit  of  the  star. 


INTERPOLATION  27 

Part  III,  "  Phenomena,"  is  of  no  direct  use  in  practical  field 
astronomy. 

Following  Part  III  is  a  table  giving  the  latitude  and  longitude 
of  (in  1916)  252  places  on  the  earth's  surface,  and  tables  num- 
bered I  to  VII,  for  the  convenient  conversion  of  units,  etc. 
Following  the  tables,  and  immediately  preceding  the  general 
index,  is  a  useful  "  Index  to  Apparent  Places  of  Stars." 

A  comparatively  small,  paper-covered  book,  called  the  " Ameri- 
can Nautical  Almanac,"  is  also  published  by  the  Nautical 
Almanac  Office.  (See  note  on  page  28.)  It  contains,  arranged 
in  slightly  different  form  from  that  in  which  the  same  data 
are  given  in  the  larger  book,  tables  which  are  sufficient  for  the 
work  of  practical  field  astronomy;  though  it  is  often  convenient 
in  preparing  for  observations  for  time  to  have  the  longer  lists 
of  stars  from  which  to  choose.  The  arrangement  and  use  of 
the  tables  will  be  understood  upon  examination,  and  need  no 
explanation  here. 

Reprints  of  portions  of  the  "  Nautical  Almanac  "  which  are 
published  by  the  different  instrument-makers  are  useful  chiefly 
in  connection  with  the  solar  attachments  for  transits.  The  use 
of  these  attachments  is  discussed  briefly  in  Appendix  B. 

23.  Interpolation.  If  the  right  ascension  or  the  declination 
of  the  sun  or  the  equation  of  time  is  desired  for  a  given  instant 
of  local  mean  time  it  is  necessary  to  add  (algebraically)  to  the 
corresponding  quantity  given  in  the  tables  for  the  instant  of 
Greenwich  mean  noon  preceding,  the  hourly  change  or  variation 
multiplied  by  the  number  of  hours  that  have  elapsed  between 
the  instant  of  Greenwich  mean  noon  and  the  instant  for  which 
the  quantity  is  desired.  The  interval  between  Greenwich  noon 
and  local  noon  is  equal  to  the  longitude  of  the  place  expressed 
in  units  of  time.  The  interval  between  Greenwich  noon  and 
noon  by  standard  time  is  equal  to  the  longitude  of  the  standard 
meridian,  and  therefore  to  some  number  of  whole  hours. 

If  the  sun's  right  ascension  or  declination  or  the  equation 
of  time  at  some  instant  of  local  apparent  time  is  desired  it  may 
be  obtained  most  exactly  from  the  ephemeris  for  Washington 
apparent  noon  in  a  manner  similar  to  that  outlined  above.  The 
interval  between  Washington  noon  and  local  noon  is  equal  to 
the  difference,  expressed  in  units  of  time,  between  the  longi- 
tude of  Washington  (5h  08m  15S.78  west  of  Greenwich)  and  the 
longitude  of  the  observer. 

The  work  of  making  these  interpolations  is  illustrated  in  the 


28    AMERICAN   EPHEMERIS   AND   NAUTICAL   ALMAMAC 

• 

course  of  the  solution  of  the  problems  in  conversion  of  thxe, 
given  in  Chapter  V. 

It  is  usually  assumed  that  the  rate  of  variation  of  any  quantity 
is  constant  between  any  two  tabular  values.  This  is  not  always 
quite  true,  however;  and  the  variations  per  hour  are  not,  in 
general,  tabular  differences  but  rates  of  change  for  the  instants 
for  which  they  are  given — differential  coefficients.  Somewhat 
more  precise  results  may  therefore  be  obtained  by  interpolating 
from  the  nearer  of  two  tabular  quantities. 

NOTE. — The  "American  Ephemeris  and  Nautical  Almanac"  and  thd  "Amer- 
ican Nautical  Almanac"  are  two  of  the  "Astronomical  Papers"  which  in  turn 
constitute  a  part  of  the  Public  Documents  of  the  United  States.  As  is  true  of 
the  greater  number  of  Public  Documents,  they  may  be  obtained  from  the 
Superintendent  of  Documents,  payment  in  advance  being  required.  The 
price  charged  is  only  enough  to  cover  the  cost  of  printing,  binding,  paper,  etc. 
The  "American  Ephemeris  and  Nautical  Almanac"  is  sold  for  one  dollar,  and 
the  "American  Nautical  Almanac"  for  thirty  cents.  Price  List  57,  giving  the 
titles  and  prices  of  all  the  Astronomical  Papers,  will  be  sent  free  on  request  by 
the  Superintendent  of  Documents. 

The  following  instructions  are  copied  from  that  list: 

"  Remittances  should  be  made  to  the  Superintendent  of  Documents,  Govern- 
ment Printing  Office,  Washington,  D.  C.,  by  postal  money-order,  express-order, 
or  New  York  draft.  If  currency  is  sent,  it  will  be  at  sender's  risk. 

"  Postage  stamps,  coins  defaced  or  worn  smooth,  foreign  money,  and  uncer- 
tified checks  will  not  be  accepted. 

"No  charge  is  made  for  postage  on  documents  forwarded  to  points  in  the 
United  States,  Alaska,  Guam,  Hawaii,  Philippine  Islands,  Porto  Rico,  Samoa, 
or  to  Canada,  Cuba,  Mexico,  or  Shanghai.  To  other  countries  the  regular 
rate  of  postage  is  charged.** 


CHAPTER  V 

PROBLEMS  IN  CONVERSION  OF  TIME 

24.  To  Change  Local  Mean  to  Local  Apparent  Time. 

Problem:  What  was  the  local  apparent  time  at  a  place 
whose  longitude  is  91°  31'  30"  W  when  the  local  mean  time  was 
8  o'clock  A.M.,  on  March  4,  1916? 

Read  Art.  16,  page  17. 

Solution:  We  must  first  find  the  equation  of  time,  E,  for 
the  instant  of  8:00:00.0  A.M.,  LM  T  in  longitude  91°  31'  30"  W 
on  March  4.  The  first  step  is  to  change  the  civil  time  to  astro- 
nomical time.  Using  the  rules  given  in  Art.  17,  page  19,  it  is 
found  that  the  instant  of  astronomical  time  corresponding  to 
March  4,  8:00:00.0  A.M.  is  March  3,  20:00:00.0.  The  second 
step  is  to  find  the  interval  that  has  elapsed  since  the  last  pre- 
ceding mean  noon  at  Greenwich.  Twenty  hours  have  elapsed 
since  mean  noon  of  March  3  at  a  place  6*06 <n06«(91°31'30" -^15) 
west  of  Greenwich;  and  therefore  26 h  06 m  06"  have  elapsed 
since  mean  noon  of  March  3  at  Greenwich,  or  2 h  06 m  06s  since 
mean  noon  of  March  4. 

The  equation  of  time  for  the  instant  of  mean  noon  of  March 
4  at  Greenwich  (taken  from  page  4  of  the  "American  Ephemeris 
and  Nautical  Almanac")  is  11 m  53s.64;  and  the  negative  sign 
indicates  that  the  difference  between  apparent  and  mean  time 
is  negative;  that  is,  that  the  amount  of  the  equation  of  time 
must  be  subtracted  from  mean  time  to  obtain  apparent  time. 
Comparing  with  the  value  for  the  preceding  day,  we  see  that  the 
equation  of  time  is  decreasing,  numerically;  so  that  the  product 
of  the  hourly  variation,  0«.546,  by  2.10  hours  since  noon  is  to 
be  subtracted  from  the  value  at  noon. 

The  equation  of  time  for  the  instant  at  which  the  change 
is  to  be  made,  then, is:  11-  53«.64  -  2.10  X  0*.546  =  11-  52«.5, 
to  be  subtracted  from  mean  time.  The  local  apparent  time  is, 
therefore,  20^  00-  00*. 0  -  11-  52*.5  =  19*  48-  07*.5,  on 
March  3,  1916. 

The  following  is  a  convenient  form  for  the  solution  of  this 
problem : 

29 


30  PROBLEMS   IN    CONVERSION    OF   TIME 

1916 

LMT,  91:31:30  W,  civil,  March  4 A.M.         8^  00<*  OO'.O 

LMT,  91:31:30  W,  astronomical,  March  3 20    00    00.0 

Longitude  west  of  Greenwich 6  06  06 


Since  Greenwich  mean  noon,  Mar.  4 .     2  06  06 

E  =  -  (llm  53s.64  -  2.10  X  0*.546)  = -         11     52  .5 


LAT,  91:31:30  W,  March  3 19»>  48 »  07*.5 

25.  To  Change  Local  Apparent  to  Local  Mean  Time. 

Problem:  What  was  the  local  mean  time  at  a  place  whose 
longitude  is  91°  31' 30"  W  when  the  local  apparent  time  was 
17"  46*  09".4  on  March  8,  1916? 

Read  Art.  16,  page  17. 

Solution:  The  same  process  is  followed  as  in  the  preceding 
problem  except  that  the  equation  of  time  may  be  taken  from 
the  Washington  ephemeris,  where  it  is  given  for  the  instant  of 
Washington  apparent  noon  of  each  day. 

The  longitude  west  of  Greenwich,  6h  06 m  06",  minus  5h  08  ^ 
15  8.78  (the  longitude  of  Washington  west  of  Greenwich)  gives 
the  longitude  of  the  place  as  Oh  57  m  50s.2  west  of  Washington. 
The  equation  of  time  for  an  instant  which  is  17h  46m  09s.4  +  00h 
57*>50s.2  =  18^43^598.6  after  the  instant  of  Washington  ap- 
parent noon  on  March  8  may  best  be  obtained  from  the  value  of 
the  equation  of  time  for  March  9,  given  on  page  515  of  the 
"American  Ephemeris  and  Nautical  Almanac."  (We  interpolate 
from  the  value  for  March  9  rather  than  from  the  one  for  March  8 
because  the  former  is  the  nearer.)  It  will  be  equal  to  10m  39s.35 
+  5.27  X  0".636  =  10m  42<».7;  and  the  positive  sign  indicates  that 
the  difference  between  mean  and  apparent  time  is  positive, 
that  is,  that  the  amount  of  the  equation  of  time  must  be  added 
to  apparent  time  to  obtain  mean  time.  (This  relation  is  shown 
by  the  heading  of  the  column:  "  Mean  —  App.") 

Had  we  used  the  Greenwich  ephemeris  and  assumed  the  value 
of  the  equation  of  time  as  given  for  mean  noon  of  March  9  to 
be  the  proper  value  for  apparent  noon  of  the  same  day,  we  should 
have  obtained  (following  the  method  of  the  preceding  article 
and  interpolating  from  the  nearer  of  the  two  values)  the  value 
of  the  equation  of  time  to  be:  10*>  42-.S.  In  this  case  the 
error  is  0".l,  and  in  no  case  is  it  likely  to  be  greater  than  about 
08.2 — an  amount  that  is  of  little  account  in  work  done  with 
ordinary  field  instruments,  This  error  could  be  reduced  by  a 


TO   CHANGE   STANDARD   TO   LOCAL   MEAN  31 

second  computation  to  an  amount  negligible  in  any  work  of 
field  astronomy. 

In  the  "American  Nautical  Almanac"  (the  small  paper- 
covered  edition)  no  ephemeris  for  apparent  noon  for  any  place 
is  given,  but  it  is  seen  from  the  example  above  that  the  one  for 
mean  noon  may  be  used  as  for  apparent  noon  without  appre- 
ciable error. 

The  following  form  of  computation  is  convenient  for  the 
solution  of  the  problem  above: 
1916 

LAT,  91:31:30  W,  March  8 17"  46*  09*.4 

Longitude  west  of  Washington.  .  .   00  57  50.2 
Since  Wash.  app.  noon,  Mar.  8.  .  .    18  43  59.6 

E  =  +  (10m  39*.35  +  5.27  X  0*.636)  = +         10    42.7 

LMT,  91:31:30  W,  astronomical,  March  8 17*  56™  52U 

LMT,  91:31:30  W,  civil,  March  9 A.M.         5    56    52  .1 

26.  To  Change  Standard  to  Local  Mean  Time. 

Problem :  What  is  the  local  mean  time  in  longitude  88°  30'  00" 
W  when  the  standard  time  is  7:31:15  A.M.? 

Read  Art.  18,  page  20. 

Solution :  The  meridian  at  which  local  mean  time  is  required 
is  1°  30'  00"  east  of  the  standard  meridian  for  the  Central  time 
belt.  Therefore,  following  the  rules  of  Art.  18,  we  must  add 
the  difference  in  longitude,  expressed  in  units  of  time,  to  standard 
time  to  get  local  mean  time. 

Std.  T  (Central) ,  .A.M.         7*  31 »  15- 

Longitude  correction,  standard  to  local +         6    00 

LMT,  88:30:00  W A.M.         7*  37-  15" 

27.  To  Change  Local  Mean  to  Standard  Time. 

Problem:  What  is  the  standard  (Central)  time  in  longitude 
91°  31'  30"  W  when  the  local  mean  time  is  8:15:27  P.M.? 

Solution:  The  meridian  at  which  the  local  mean  time  is 
given  is  1°  31'  30",  or  Oh  06 m  06 «,  west  of  the  standard  meridian 
for  the  Central  time  belt.  Therefore,  following  the  rules  of 
Art.  18,  we  must  add  the  difference  in  longitude,  expressed  in 
units  of  time,  to  the  local  mean  time  to  get  standard  time. 

LMT,  91:31:30  W P.M.          8*  15-  27* 

Longitude  correction,  local  to  standard +         6    06 

Standard  time  (Central) P.M.          8*  21 «  33* 


32  PROBLEMS   IN   CONVERSION   OF  TIME 

28.  To  Change  Standard  to  Local  Apparent  Time. 

Problem:  What  was  the  local  apparent  time  at  a  place 
whose  longitude  is  91°  31'  30"  W  when  it  was  8:00:00.0  A.M. 
by  standard  time  (Central),  on  August  27,  1915? 

Read  Arts.  16  to  18,  pages  17  to  20. 

Solution:  The  procedure  is  almost  exactly  that  used  in 
Art.  24,  page  29.  The  reason  for  the  slight  difference  will  be 
self-evident  if  we  remember  that  standard  time  in  longitude 
91°  31'  30"  W  is  simply  the  local  mean  time  at  longitude 
90°  00'  00"  W. 

1915 

Std.  T,  91:31:30  W,  civil,  August  27 A.M.         8»>  00™  (XKO 

LMT,  90:00:00  W,  astronomical,  August  26.  .  20  00  00.0 
LMT,  91:31:30  W,  astronomical,  August  26.  .  19  53  54  .0 
Longitude  west  of  Greenwich 6  06  06 


Since  Greenwich  mean  noon,  Aug.  27 .  2  00  00 

E  =  -  (1m  42'.02  -  2.00  X  0«.704)  = -  1    40  .6 


LAT,  91:31:30  W,  August  26 19 fa  52 ^  13-.4 

29.  To  Change  Local  Apparent  to  Standard  Time. 

Problem:  What  was  the  standard  (Central)  time  when  the 
local  apparent  time  at  a  place  whose  longitude  is  91°  31'  30"  W, 
was  16fa  08^  17s.4  on  February  2,  1916? 

Read  Arts.  16  to  18,  pages  17  and  20. 

Solution:  The  procedure  is  just  the  reverse  of  that  of  the 
preceding  article,  with  the  exception  that  the  equation  of  time 
is  obtained  from  the  ephemeris  of  the  sun  for  Washington  appar- 
ent noon.  Interpolation  is  made  from  the  nearer  of  two  tabular 
values  of  the  equation  of  time. 

1916 

LAT7,  91:31:30  W,  February  2 16"  08*  17-.4 

Longitude  west  of  Washington. ...     0  57  50 . 2 


Since  Wash.  app.  noon,  Feb.  2.  .  .   17  06  07.6 

E  =  +  (13-  53«.48  -  6.90  X  0*.297)  = -f         13    51  .4 


LMT,  91 :31 :30  W,  astronomical,  February  2. .  16^  22-  08*.8 
LMT,  90:00:00  W,  astronomical,  February  2. .  16  28  14  .8 
Std.  T,  Central,  February  3 AM.  4  28  14  .8 


TO   CHANGE  LOCAL  MEAN   SOLAR  TO   SIDEREAL      33 

30.  To  Change  Local  Mean  Solar  to  Sidereal  Time. 

Problem:  What  was  the  sidereal  time  at  6:00:00.0  A.M., 
local  mean  time,  in  longitude  91°  31'  30"  W  on  October  20,  1915? 

Read  Art.  21,  page  23. 

Solution:  Since  the  increase  in  the  value  of  the  right  ascen- 
sion of  the  mean  sun  during  any  interval  is  equal  to  the  gain  of 
the  apparent  movement  of  the  vernal  equinox  over  that  of  the 
mean  sun  during  that  interval — i.e.,  it  is  equal  to  the  difference 
between  the  number  of  sidereal  units  and  the  number  of  solar 
units  in  the  interval — it  is  convenient  in  working  problems  to 
apply  this  increase  in  the  following  manner: 

Determine  the  value  of  the  right  ascension  of  the  mean  sun 
for  the  instant  of  local  mean  noon.  This  quantity  will  be 
called  An.  It  may  be  found  by  adding  to  the  value  of  the  sun's 
right  ascension,  A,  for  the  instant  of  Greenwich  mean  noon  of 
the  same  date  (i.e.,  in  west  longitudes)  the  increase  in  right 
ascension  during  a  solar  interval  equal  to  the  longitude  of  the 
place.  This  increase  may  be  taken  directly  from  Table  II 
at  the  back  of  this  book  or  from  Table  III  in  the  "  Nautical 
Almanac."  For  any  given  longitude  this  increase  or  correction 
for  reducing  the  sun's  right  ascension  at  Greenwich  mean  noon 
to  its  value  for  local  mean  noon  is  a  constant  which  may  be 
used  for  all  similar  problems.  It  is  the  amount  by  which  the 
distance  traveled  by  the  vernal  equinox  (expressed  in  hours) 
exceeds  the  distance  traveled  by  the  sun  during  the  interval  that 
the  sun  has  occupied  in  coming  from  the  Greenwich  meridian 
to  the  local  meridian. 

To  this  value  of  AM  add  the  local  mean  time  expressed  in 
sidereal  units,  i.e.,  the  sidereal  interval  since  mean  noon — the 
distance  expressed  in  hours  that  the  vernal  equinox  has  traveled 
since  mean  noon.  This  change  from  solar  to  sidereal  units 
may  also  be  made  by  aid  of  Table  II. 

This  sum — the  right  ascension  of  the  sun  at  local  mean  noon 
plus  the  sidereal  interval  since  local  mean  noon — will  be  the 
sidereal  time. 

For  the  problem  above:  First  change  6:00:00.0  A.M., 
October  20  to  18*  00™  00".0,  October  19.  The  value  of  A  for 
Greenwich  mean  noon  of  October  19  is  (from  the  "American 
Ephemeris  and  Nautical  Almanac")  13h  47™  309.95.  The  in- 
crease for  6h  06™  06"  west  longitude  is,  from  Table  II,  1*>  00*.15 
(or,  from  Table  III  of  the  "  Nautical  Almanac,"  lmOO<>.140), 
making  An  equal  to  13*  48™  31*.l.  18*00™  00«.0  LMT  ex- 


34  PROBLEMS   IN   CONVERSION   OF  TIME 

pressed  in  sidereal  units  isjfrom  Table  II,  or  Table  III  of  the 
"Nautical  Almanac")  18*  02*>  57*.4.  This  is  the  sidereal  in- 
terval since  local  mean  noon,  or  the  number  of  hours  of  arc  that 
the  vernal  equinox  covered  while  the  mean  sun  was  passing  over 
18 fc  00*>  (XKO  of  arc. 

The  sidereal  time  is  therefore  13^  48™  3K1  plus  18 *  02m  57  s.^ 
or  (after  subtracting  24  hours)  7h  51 m  288.5. 

The  following  form  of  solution  may  be  used: 

1915 

LMT,  91:31:30  W,  civil,  October  20 A.M.  6*  00™  00«.0 

LMT,  91:31:30  W,  astronomical,  October  19.  .  18    00    00  .0 

Correction,  solar  to  sidereal +           2    57  .4 


Sidereal  interval  since  mean  noon 18  »>  02™  57*.4 

A,    at    Greenwich    mean    noon, 

October  19 13  47  30.95 

Increase  due  to  longitude 1  00 . 15 


An 13    48    31.1 

Sidereal  time  on  October  19,  solar  (astr.)  date.        SI*  51™  28°.5 

or         7    51    28.5 

31.    To  Change  Sidereal  to  Local  Mean  Solar  Time. 

Problem:  What  was  the  local  mean  time  on  October  19,  1915, 
(astronomical  date)  in  longitude  91°  31'  30"  W  when  the  sidereal 
time  was  7>>30™  27  °.5? 

Read  Art.  21,  page  23. 

Solution:  The  solution  is  much  the  same — except  that  the 
order  is  reversed — as  that  of  the  preceding  problem,  the  increase 
in  the  right  ascension  of  the  mean  sun  being  taken  care  of  in 
the  same  way. 

From  the  given  sidereal  time  subtract  An  (first  adding  24 
hours  to  the  sidereal  time  if  necessary  for  the  subtraction).  The 
difference  is  the  sidereal  interval  since  local  mean  noon.  Change 
this  sidereal  interval  to  the  corresponding  solar  interval  by  use 
of  Table  I  (or  Table  II  of  the  "  Nautical  Almanac")  and  the 
result  is  local  mean  time. 

Stated  in  other  words,  the  result  of  the  subtraction  is  the 
number  of  hours  of  arc  which  the  vernal  equinox  has  covered 
since  local  mean  noon.  From  this  result  is  computed  the  number 


TO   CHANGE   SIDEREAL   TO  LOCAL   MEAN   SOLAR      35 

of  hours  of  arc  which  the  mean  sun  has  covered  since  local  mean 
noon;  i.e.,  the  local  mean  time. 

It  should  be  clearly  understood  that  adding  24  hours  to,  or 
subtracting  24  hours  from,  the  hour  angle  of  the  vernal  equinox 
does  not  in  any  way  affect  the  date;  for  the  date  bears  no  relation 
to  the  hour  angle  of  the  vernal  equinox — i.e.,  to  the  sidereal  time 
— but  is  dependent  only  on  the  sun.  In  other  words,  dates  are 
always  solar  dates. 

The  following  form  of  solution  may  be  used  for  the  problem 
above: 

1915 

Sidereal  time,  91:31:30  W,  Oct.  19,  astr.  date. .          7*>  30™  27".5 
A,    at    Greenwich    mean    noon, 

Oct.  19 13  47  30.95 

Increase  due  to  longitude 100. 15 


An 13    48    31  .1 


Sidereal  interval  since  mean  noon 17 h  41  ^  568.4 

Correction,  sidereal  to  solar —  2    54  .0 


LMT,  astronomical,  91:31:30  W,  Oct.  19 17"  39*  02-.4 

LMT,  civil,  91:31:30  W,  Oct.  20 A.M.         5   39    02  .4 


CHAPTER  VI 
OBSERVATIONS— CORRECTIONS  TO  OBSERVATIONS 

32.  Objects  Observed.    Methods  of  Naming  Stars.    Obser- 
vations are  to  be  made  on  the  sun  and  on  the  stars.     The  moon 
and  the  planets  are  sometimes  used  as  objects  for  observations, 
especially  for  longitude;    but  these  observations  more  properly 
form  a  part  of  geodetic  work  of  greater  refinement  than  is  herein 
contemplated,  and  their  discussion  will  be  omitted. 

The  stars  are  distinguished  according  to  the  following  scheme : 
The  sky  is  divided  into  irregular  areas,  usually  such  that  the 
stars  in  a  given  division  seem  to  form  a  natural  group;  and  all 
the  stars  within  that  area  form  a  constellation,  which  receives  a 
name.  The  individual  stars  of  a  constellation  are  sometimes 
distinguished  by  receiving  a  special  name,  and  usually  by  a 
Greek  letter  or  a  number  also.  The  letters  of  the  Greek  alphabet 
are  usually  assigned  to  the  stars  of  a  constellation  in  descending 
order  of  brightness;  a.  to  the  brightest,  /3  to  the  next,  and  so 
on.  A  star  is  then  named  by  stating  its  letter  followed  by  the 
name  of  the  constellation  to  which  it  belongs  in  the  Latin  genitive 
form.  Thus  the  "pole-star"  has  the  special  name  "  Polaris," 
and  since  it  is  the  brightest  star  in  the  constellation  "  Ursa 
Minor,"  it  is  also  called  "a  Ursse  Minoris."  Sometimes  two 
stars  which  are  apparently  very  close  together  are  given  the 
same  letter;  in  which  case  a  small  number  is  placed  over  their 
letter,  as  a1,  a2,  etc.,  to  distinguish  them  and  to  indicate  the  order 
in  which  they  cross  the  meridian. 

The  brightness  of  a  star  is  designated  by  a  number  from  a 
numerical  scale  of  magnitudes.  In  this  scale  the  numbers  in- 
crease as  the  magnitudes  decrease.  Stars  of  the  fifth  magnitude 
are  about  as  dim  as  can  be  seen  under  favorable  conditions  with 
the  naked  eye.  Polaris  is  of  the  second  magnitude. 

33.  Circumpolar   Constellations.     Stars   whose  co-declination 
or  polar  distance  is  less  than  the  latitude  of  the  observer  do 
not  go  below  his  horizon  at  any  point  in  their  diurnal  circles, 
and    are    called    circumpolar    stars.     Those    circumpolar    stars 
which  are  near  the  pole  are  of  most  importance  to  the  surveyor. 


CIECTJMPOLAR    CONSTELLATIONS 


37 


1 


Ursa  Major 


(Mizar) 


Ursa 
Minor 


Poleo 


"^(Polaris) 


Some  of  the  more  im- 
portant of  these  are 
here  shown  in  Fig.  12. 

The  one  most  used  of 
all  because  it  is  the 
nearest  the  pole  is  the 
brightest  star  in  the 
constellation  "Ursa 
Minor,"  or  the  "  Little 
Dipper."  It  is  Polaris, 
or  a  Ursse  Minoris.  It 
is  about  1  °  08'  from  the 
pole  (1916)  and  is  ap- 
proaching the  pole  at 
the  rate  of  about  0'.3 
per  year.  There  is  no 
star  exactly  at  the  pole. 
No  other  bright  star  is 
near  Polaris  which  is 
likely  to  be  confused 
with  it;  and  it  is  easy 
to  find  by  reference  to 
two  stars  in  the  constella- 
tion Ursa  Major,  or  the 
"  Great  Dipper,"  on  the 
opposite  side  of  the  pole. 
The  two  brightest  stars 
in  this  constellation,  the 
ones  which  form  the 
side  of  the  "bowl  "  op- 
posite the  "handle"  of 
the  Dipper,  are  called 
the  "  Pointers."  A  line 
drawn  through  them  and 
produced  falls  very  near 
Polaris. 

The  star  f  Ursse  Ma- 
joris,  at  the  bend  in  the 
handle  of  the  Great  Dipper  (see  Fig.  12),  is  of  some  use  to  the  sur- 
veyor because  it  falls  very  nearly  on  the  same  hour  circle  as  Polaris 
and  5  Cassiopeise.  Cassiopeia  is  a  constellation  on  the  opposite 
side  of  Polaris  from  Ursa  Major,  the  five  brightest  stars  of  which 


Cassiopeia 


Polaris  at  Lower  Culmination. 
FIG.  12. 

SOME  OF  THE  ClRCUMPOLAR 

CONSTELLATIONS 
(At  the  North  Celestial  Pole.) 


38  OBSERVATIONS — CORRECTIONS  TO  OBSERVATIONS 

form  a  rather  awkward  "  W".  5  Cassiopeise  is  at  the  lower  left- 
hand  corner  of  the  W.  The  position  of  Polaris  in  its  diurnal 
path  around  the  pole  may  be  estimated  quite  accurately  by  the 
relative  positions  of  these  three  stars:  Polaris,  Ursa3  Majoris, 
and  5  Cassiopeise.  If  they  are  in  a  vertical  line  with  5  Cas- 
siopeise above,  Polaris  is  at  upper  culmination;  if  £  Ursse 
Majoris  is  above,  Polaris  is  at  lower  culmination.  If  they  are 
in  a  horizontal  position  with  5  Cassiopeise  at  the  right,  or  east, 
Polaris  is  at  eastern  elongation;  while  a  reversed  position  in- 
dicates western  elongation  of  Polaris. 

j8  Cassiopeise,  at  the  upper  right-hand  corner  of  the  TF,  has 
a  right  ascension  very  nearly  equal  to  zero;  i.e.,  it  is  on  an 
hour  circle  which  passes  very  near  the  vernal  equinox.  There- 
fore, the  hour  angle  of  this  star  is  closely  equal  to  local  sidereal 
time.  This  hour  angle,  and  thereby  local  sidereal  time,  may 
be  estimated  fairly  well  by  remembering  that  when  the  star  is 
vertically  above  Polaris  it  is  practically  0  hours,  when  it  i? 
vertically  below  it  is  12  hours;  and  that  the  points  half-way 
between  these  two  positions  on  the  left  and  right  correspond  to 
6  hours  and  18  hours,  respectively. 

In  the  determination  of  latitude  and  of  time  we  shall  find  use 
for  some  of  the  stars  in  other  constellations  than  those  men- 
tioned, but  we  can  identify  them  as  needed  by  means  of  their 
co-ordinates  (taken  from  the  "American  Ephemeris  and  Nautical 
Almanac,"  in  terms  of  right  ascension  and  declination,  and  con- 
verted into  other  co-ordinates  for  use)  and  they  will  not  be 
discussed  further  at  present. 

34.  Parallax.  The  co-ordinates  of  a  celestial  object  should  be 
referred  to  the  center  of  the  celestial  sphere  as  the  pole  or  origin. 
This  center  is  at  the  center  of  the  earth;  and  an  altitude  of  any 
body  nearer  than  the  fixed  stars — such  as  the  sun — which  has 
been  measured  from  the  surface  of  the  earth,  is  less  than  the 
altitude  referred  to  the  center  of  the  earth  as  the  origin  of  co- 
ordinates by  an  amount  called  the  parallax,  and  must  be  reduced 
to  the  value  for  the  center  by  applying  a  parallax  correction.  If 
the  earth  is  assumed  to  be  a  sphere,  wrhich  is  sufficiently 
accurate  for  practical  purposes,  the  effect  of  parallax  is 
to  decrease  the  altitude  of  a  body  without  affecting  the 
azimuth. 

Referring  to  Fig.  13,  the  angle  H'AS  is  the  measured  altitude 
of  a  point,  S,  obtained  by  an  observation  from  A,  a  point  on  the 
surface  of  the  earth;  and  HOS  is  its  altitude  as  referred  to  the 


PARALLAX 


39 


center  of  the  earth,  O.     The  difference  between  the  two,  or 
the  angle  ASO,  is  the  parallax  correction. 
In  the  triangle  AOS, 

OA 

sin  ASO  =  sin  OAS  —     .      .      .      .        (a) 
Go 

where  the  angle  OAS  is  equal  to  90°  plus  the  measured  altitude, 

tz 


FIG.  13 

OA  is  the  radius  of  the  earth,  OS  is  the  distance  from  the  center 
of  the  earth  to  the  center  of  the  body  observed. 

When  the  observed  body  is  at  the  zenith  it  is  evident  that 
the  parallax  will  be  zero;  and  when  it  is  on  the  horizon,  as 
observed  from  the  surface  of  the  earth  (i.e.,  at  S'),  its  parallax 
will  be  a  maximum.  This  maximum  is  called  the  horizontal 
parallax. 

For  this  case,  where  angle  OAS'  is  equal  to  90°: 


OA 
— 


(b) 


Let: 


cp  =  parallax  at  any  position  of  the  body. 
Cp  =  horizontal  parallax. 
h'  =  measured  altitude. 
Since  OS  =  OS',  Equation  (b)  may  be  written: 

OA 

smC,  =  —  . 


40  OBSERVATIONS — CORRECTIONS  TO  OBSERVATIONS 

Substituting  this  value  for  OA/OS  in  Equation  (a),  and 
remembering  that  sin  OAS  =  cos  h'y  we  obtain: 

sin  Cp  =  sin  Cp  .  cos  In!  (c) 

Since  cp  and  CP  are  very  small  angles  we  may  without  ap- 
preciable error  substitute  the  angles  for  their  sines,  and  Equation 
(c)  then  becomes: 

c"p  =  C"p  .  cosh'    .      .     .     .     .      (21) 

in  which  cp  and  CP  are  both  expressed  in  seconds  of  arc. 

For  the  sun  the  mean  value  of  Cp  is  8".8.  For  the  moon 
and  for  the  planets  it  is  much  larger.  For  the  stars  it  is  too 


FIG.  14. 

small  to  be  measured,  because  of  their  great  distance  from 
the  earth.  There  is,  therefore,  no  parallax  correction  to  measured 
altitudes  of  the  fixed  stars. 

The  parallax  correction,  when  needed,  is  always  to  be  added 
to  observed  altitudes. 

Parallax  corrections  to  measured  altitudes  of  the  sun  are 
given  in  Table  IV. 

35.  Refraction.  When  a  ray  of  light  passes  from  one  medium 
to  another  of  different  density  it  is  bent,  and  this  bending  is 
called  refraction.  As  a  ray  of  light  comes  from  a  celestial  body 
to  the  eye  of  an  observer  it  passes  through  successive  layers  of 
atmosphere  of  increasing  density,  and  is  therefore  bent  down- 
ward in  a  curve,  as  shown  in  Fig.  14. 


INSTRUMENTAL   ERRORS  41 

If  an  observer  on  the  surface  of  the  earth,  at  0,  is  looking  at 
a  star  which  is  actually  at  S,  it  will  appear  to  him  to  be  at  some 
point  above  $,  as  at  $';  OS'  being  a  tangent  at  0  to  the  curve 
AO.  The  angle  which  must  be  subtracted  from  the  measured 
altitude,  HOS't  to  obtain  the  altitude  of  S  is  called  the  refrac- 
tion correction. 

A  convenient  rule  for  the  amount  of  this  correction,  derived 
by  application  of  the  laws  of  physics  after  some  simplifying 
assumptions  have  been  made,  is  that  the  refraction  correction 
in  minutes  is  equal  to  the  natural  co-tangent  of  the  observed 
altitude.  This  is  sufficiently  accurate  for  all  altitudes  greater 
than  about  ten  degrees  which  have  been  measured  with  ordinary 
field  instruments — transit  or  sextant.  Differences  in  tem- 
perature and  in  barometric  conditions  cause  differences  in  the 
amount  of  the  refraction  correction,  but  they  are  too  slight  to 
require  attention  in  work  of  this  character. 

Table  III  gives  the  amount  of  the  refraction  correction  to  be 
applied  to  observed  altitudes  for  a  mean  barometric  pressure 
and  temperature. 

The  refraction  correction  is  always  to  be  subtracted  from  an 
observed  altitude. 

36.  Semi-diameter.    The  sun's  disc,  as  seen  through  a  tele- 
scope, is  circular,  and  measurements  are  usually  made  to    its 
edge,  or  limb,  rather  than  to  its  center.     Such  measurements 
must  be  corrected  by  the  amount  of  the  angle  subtended  by  the 
semi-diameter  of  the  sun  to  reduce  them  to  the  proper  values 
for  the  center.     As  has  been  mentioned,  the  amount  of  this 
angular  semi-diameter  is  given  for  each  day  in  the  year  in  the 
"Nautical  Almanac";    and  may  be  taken  therefrom  for  use 
in  reducing  observations. 

Values  of  the  sun's  semi-diameter  for  the  first  of  each  month 
in  the  year  are  given  in  Table  IV  at  the  back  of  this  book.  Inter- 
polations from  this  table  are  sufficiently  accurate  for  most  work 
with  field  instruments. 

37.  Instrumental  Errors.    Though  it  would  be  well  if  an 
instrument  in  perfect  adjustment  could  be  used  for  all  astro- 
nomical observations,  it  is  not  always  possible;  and  care  should 

taken  to  eliminate,  so  far  as  may  be,  the  effect  of  any  in- 

of  adjustment  or  construction. 
Considering  first  the  transit:    When  it  is  in  use  care  should 
be  taken  to  make  the  plates  truly  horizontal.     If  the  plate 
bubbles  are  not  in  good  adjustment  this  can  still  be  accomplished 


42  OBSERVATIONS — CORRECTIONS  TO  OBSERVATIONS 

by  leveling  up,  turning  the  plates  180°,  and  by  means  of  the 
leveling  screws  bringing  each  bubble  half-way  back  to  the  center 
of  its  tube.  When  the  plates  are  level  both  bubbles  should 
remain  in  the  same  positions  in  the  tubes  throughout  a  complete 
revolution  of  the  plates. 

Whenever  possible  horizontal  angles  should  be  repeated — at 
least  doubled — making  half  of  the  measurements  with  the 
telescope  direct  and  half  with  it  inverted.  The  average  of  the 
results  is  then  free  from  the  effect  of  errors  in  the  line  of  sight 
or  in  the  height  of  standards,  and  should  be  of  greater  precision 
than  any  one  reading. 

Vertical  angles  cannot  be  repeated;  but  whenever  time  permits 
at  least  two  readings  should  be  taken,  one  with  the  telescope 
direct  and  one  with  it  inverted,  releveling  if  necessary  after 
reversing  the  instrument.  This,  of  course,  presupposes  the  use 
of  a  transit  with  a  full  vertical  circle.  The  mean  of  the  two 
readings  should  be  free  from  the  effect  of  errors  in  adjustment 
in  line  of  sight,  standards,  telescope  level,  and  index  error.  If, 
because  the  instrument  does  not  have  a  full  vertical  circle  or 
because  of  lack  of  time,  the  above  method  cannot  be  used,  care 
must  be  taken  to  see  that  the  axis  of  the  telescope  level  is  parallel 
to  the  line  of  sight;  and  the  index  error  must  be  determined  and 
proper  correction  made.  The  index  error  is  the  reading  of  the 
vertical  arc  or  circle  when  the  line  of  sight  is  horizontal. 

It  may  be  convenient  to  remember  that  for  use  with  vertical 
angles  read  above  the  horizontal,  if  the  zero  of  the  vernier  is  to 
the  right  of  the  zero  of  the  vertical  arc  when  the  telescope  is 
horizontal  the  index  correction  is  positive;  while  if  the  zero  of 
the  vernier  is  to  the  left  of  the  zero  of  the  vertical  arc  when  the 
telescope  is  horizontal  the  index  correction  is  negative. 

Concerning  the  sextant:  The  sextant  is  not  adapted  to  the 
measurement  of  horizontal  angles  between  objects  at  different 
elevations;  but  vertical  angles  may  often  be  measured  with 
greater  precision  with  the  sextant  than  with  the  transit,  on 
account  of  the  greater  radius  and  finer  graduation  of  the  limb. 
There  is  usually  an  index  error,  whose  amount  should  be  deter- 
mined and  the  readings  properly  corrected. 

The  amount  and  sign  of  the  index  error  may  be  determined 
in  the  following  manner:  Using  an  artificial  horizon,  bring 
the  direct  and  reflected  images  of  the  sun  externally  tangent  to 
each  other  in  each  of  the  two  possible  positions,  and  read  the 
vernier,  at  each  setting.  It  will  be  noticed  that  it  is  necessary 


SUGGESTIONS    FOE   OBSERVING  43 

to  consider  the  numbering  of  the  divisions  on  the  vernier  reversed 
when  making  the  reading  to  the  right  of  the  zero  of  the  main 
limb.  Call  the  reading  to  the  left  minus  and  that  to  the  right 
plus.  Half  the  algebraic  sum  of  the  two  readings  is  then  the  index 
error,  with  the  proper  algebraic  sign.  It  should  be  remembered 
that  the  index  correction  will  have  the  opposite  sign. 

38.  Sequence  of  Corrections.     Corrections  to  observed  alti- 
tudes should  be  made  in  the  following  order: 

(1)  Instrumental  corrections. 

(2)  Refraction  correction. 

(3)  Semi-diameter  correction. 

(4)  Parallax  correction. 

The  chief  and  probably  the  only  instrumental  correction  that 
can  be  applied  if  ordinary  field  instruments  are  used  will  be  the 
index  correction.  Care  should  be  taken  to  give  it  the  proper 
sign. 

The  algebraic  sum  of  the  refraction  and  parallax  corrections 
is  often  applied  as  a  single  correction. 

When  applying  a  refraction  correction  to  an  observed  altitude 
of  the  sun,  the  correction  for  the  limb  observed — not  for  the 
center — should  be  used.  Because  they  are  at  different  altitudes 
the  values  of  the  corresponding  corrections  would  differ  con- 
siderably if  the  altitudes  were  small. 

39.  Suggestions  for  Observing.     When  making  astronomical 
observations  greater  care  is  necessary  in  the  instrumental  work 
than  is  often  exercised  in  ordinary  surveying. 

Considering  first  the  use  of  the  transit: 

A  good,  -firm  "set-up"  should  be  secured;  and  the  two  plates 
of  the  leveling  head  should  be  nearly  parallel  when  the  instru- 
ment is  leveled.  The  leveling  screws  should  be  turned  to  just 
the  proper  degree  of  tightness — not  too  loose  so  as  to  allow  the 
instrument  to  rock,  nor  so  tight  as  to  bind  and  later  spring  the 
plates  from  a  horizontal  position.  So  far  as  possible  the  effect 
of  instrumental  errors  should  be  eliminated  by  the  method  of 
observing. 

Extra  care  is  needed  in  reading  angles  at  night;  that  is,  in 
determining  the  reading  of  the  vernier  at  a  given  setting.  A 
lantern — or  better,  an  electric  flash-light — should  be  held  beside 
and  rather  back  of  the  head  of  the  observer.  It  will  be  noticed 
that  there  is  greater  likelihood  of  getting  a  wrong  reading  because 
of  not  looking  squarely  down  on  the  vernier  than  by  daylight. 

When  observing  at  night  the   cross-hairs  must  usually  be 


44  OBSERVATIONS CORRECTIONS  TO  OBSERVATIONS 

illuminated  in  order  to  be  seen.  A  reflector  made  by  an  instru- 
ment-maker is  sometimes  used.  It  is  simply  a  cylinder  to  be 
put  on  the  telescope  in  place  of  the  sunshade,  from  the  side  of 
which  a  large  "notch"  has  been  cut.  This  notch  is  framed  with 
brightly  polished  metal,  so  placed  as  to  reflect  light  down  the 
barrel  of  the  telescope  from  a  lantern  held  beside  it. 

If  such  a  reflector  is  not  at  hand  one  which  will  serve  equally 
well  may  be  made  from  a  piece  of  white  paper  three  or  four 
inches  wide  and  long  enough  to  wrap  around  the  objective  end 
of  the  telescope,  where  it  is  held  in  place  by  a  rubber  band. 
A  crescent-shaped  cut  in  the  side  of  the  paper  cylinder  thus 
formed  will  produce  a  flap  which  may  be  pushed  inward  to  act 
as  the  reflector. 

Before  attempting  to  "  find  "  a  star  with  the  telescope  the  eye- 
piece should  be  properly  focused  on  the  cross-hairs  and  the 
objective  focused  on  a  distant  object,  such  as  a  distant  light, 
and  then  they  should  not  be  disturbed;  as  it  is  difficult  to  find 
a  star  if  the  telescope  is  not  properly  focused.  This  focusing, 
especially  that  of  the  eye-piece,  may  profitably  be  done  before 
dark.  The  star  may  be  found  by  sighting  first  along  the  telescope 
and  then  through  it  before  a  light  is  brought  near  the  instrument. 
When  the  star  is  clearly  visible  as  a  point  of  light  the  lantern 
should  be  gradually  brought  nearer  the  reflector  until  the  cross- 
hairs may  be  seen  distinctly,  but  not  so  close  as  to  make  the 
field  so  bright  that  the  star  cannot  be  plainly  seen.  The  star 
will  not  appear  any  larger  or  brighter  through  the  telescope 
than  when  seen  by  the  naked  eye. 

Suggestions  in  regard  to  "lining  in"  a  point  at  night,  or 
establishing  a  reference-mark  from  which  horizontal  angles 
may  be  measured  at  night,  are  given  in  connection  with  the 
work  in  which  these  operations  are  required;  namely,  at  the 
beginning  of  Chapter  VIII  on  "  Observations  for  Azimuth." 
The  methods  of  lining  in  a  stake  and  tack  serve  equally  well  to 
suggest  methods  of  sighting  on  a  stake  already  set. 

When  making  observations  on  the  sun  the  eye-piece  of  the 
telescope  must  be  covered  with  a  piece  of  dark  glass  to  protect 
the  eye.  For  measuring  altitudes  greater  than  fifty  or  sixty 
degrees  a  prismatic  eye-piece  must  be  used,  fastened  on  over 
the  regular  eye-piece.  When  using  this  attachment  in  making 
observations  on  the  sun  it  should  be  remembered  that  the  prism 
turns  the  image  upside  down,  but  not  right  for  left.  If  either 
the  colored  glass  or  the  prismatic  eye-piece  is  not  at  hand  when 


SUGGESTIONS  FOR  OBSERVING          45 

needed  the  following  scheme  may  be  used,  provided  the  telescope 
has  an  erecting  eye-piece: 

Draw  the  eye-piece  nearly  out  and  the  objective  nearly  in 
by  means  of  the  focusing  screws.  If  now  the  telescope  is  pointed 
toward  the  sun  and  a  white  card  held  below  and  three  or  four 
inches  from  the  eye-piece,  the  images  of  the  sun  and  of  the  cross- 
hairs may  both  be  focused  on  the  card  by  slight  manipulations  of 
the  focusing  screws;  and  a  pointing  may  be  made  quite  .ac- 
curately without  actually  looking  through  the  telescope.  If  the 
eye-piece  is  non-erecting  this  method  cannot  be  used. 

Concerning  the  use  of  the  sextant: 

For  the  theory,  adjustments,  and  general  method  of  use  of  the 
sextant  the  reader  is  referred  to  any  standard  text  on  surveying.* 
The  method  of  determining  the  index  error  has  been  given  in 
Art.  37,  page  42. 

As  has  been  noted,  the  sextant  is  not  adapted  to  the  measure- 
ment of  horizontal  angles  between  points  not  at  the  same  eleva- 
tion; so  that  its  use  in  the  work  described  in  this  book  will  be 
limited  to  the  observation  of  the  altitude  of  the  sun  or  of  a  star. 
Measuring  the  altitude  of  a  celestial  body  with  a  sextant  con- 
sists in  measuring  the  angle  between  the  object  itself  and  its 
reflection  from  a  surface  called  an  "artificial  horizon." 


FIG.  15. 

In  Fig.  15,  A  represents  the  position  of  the  eye  of  the  observer, 
B  the  artificial  horizon,  and  SA  and  S'B  rays  from  the  same 
celestial  object  to  the  eye  and  to  the  artificial  horizon,  respec- 
tively. S'B  is  reflected  along  the  line  BA.  A  H  is  a  horizontal 

*See,  for  instance,  Raymond's  "Plane  Surveying,"  Second  Edition,  page 
393;  or  Breed  and  Hosmer,  "Principles  and  Practice  of  Surveying,"  Volume 
II,  page  274. 


46  OBSERVATIONS CORRECTIONS  TO  OBSERVATIONS 

line  through  the  eye  of  the  observer,  and  EH'  is  a  horizontal 
line  as  determined  by  the  surface  of  the  artificial  horizon.  It 
is  evident  that  all  the  angles  marked  a  in  the  figure  are  equal; 
and  that  the  measured  angle,  SAB,  is  therefore  equal  to  twice 
the  apparent  altitude  of  the  body  S.  The  angle  is  measured 
by  observing  through  the  transparent  portion  of  the  horizon 
glass  of  the  sextant  the  image  reflected  from  the  artificial  horizon, 
and  bringing  into  coincidence  (or  tangency)  with  it  the  image 
of  the  object  as  reflected  from  the  index  glass. 

When  measuring  the  altitude  of  the  sun,  using  either  no 
telescope  or  an  erecting  telescope,  if  the  apparent  lower  limb  of 
the  sun  as  reflected  from  the  index  glass  is  brought  into  contact 
with  the  apparent  upper  image  seen  in  the  artificial  horizon,  the 
angle  measured  is  twice  the  altitude  of  the  sun's  lower  limb. 
If  the  telescope  is  an  inverting  one,  the  angle  measured  by  this 
method  is  twice  the  altitude  of  the  upper  limb.  The  index 
correction  should  be  applied  before  the  measured  angle  is  divided 
by  two  to  obtain  the  altitude. 

A  shallow  dish  of  mercury  is  usually  considered  most  satis- 
factory for  an  artificial  horizon,  but  a  dish  of  molasses  will 
answer  nearly  or  quite  as  well.  Whatever  is  used,  it  should 
be  protected  from  disturbance  from  the  wind  or  other  causes 
during  the  observations.  A  roof -shaped  cover  with  glass  windows 
is  usually  provided  for  use  with  the  mercury  horizons.  This  is 
satisfactory  if  the  two  faces  of  each  piece  of  glass  are  parallel 
planes.  A  cover  of  fine  mosquito  netting  will  serve  the  purpose 
quite  well,  and  introduce  no  error  from  refraction. 

A  good  deal  of  care  and  considerable  practice  are  required  to 
obtain  accurate  results  from  the  use  of  the  sextant;  but  because 
of  its  finer  graduation,  which  is  made  possible  on  account  of  the 
greater  radius  of  the  limb  as  compared  with  a  transit  (sextants 
are  commonly  graduated  to  read  to  the  nearest  ten  seconds,  or 
at  least  to  the  nearest  half  minute),  it  is  capable  of  giving  very 
precise  results.  Because  of  its  portability  it  is  adapted  for  use 
in  places  where  a  transit  could  not  be  used.  Since  sights  to 
both  objects  which  determine  the  angle  to  be  measured  are 
taken  at  the  same  time  it  is  not  necessary  that  the  sextant  have 
a  firm  support,  such  as  is  required  for  the  transit.  It  is  the 
instrument  used  for  astronomical  observations  at  sea;  in  which 
case  the  sea  horizon  instead  of  an  artificial  horizon  is  used  from 
which  to  measure  altitudes,  requiring  a  correction  for  the  "dip" 
of  the  apparent  below  the  true  horizon. 


SUGGESTIONS   FOR   OBSERVING  47 

There  is  usually  furnished  with  the  sextant  a  special  telescope 
for  astronomical  work.  There  are  several  colored  glasses  at- 
tached to  the  frame  which  may  be  turned  into  the  line  of  sight 
to  protect  the  eye  when  the  sun  is  the  object  observed.  It  is  a 
good  plan  to  use  one  color  in  front  of  the  horizon  glass  and  a 
different  color  in  front  of  the  index  glass,  so  that  the  two  images 
may  be  of  different  color  and  be  more  easily  distinguished. 

One  final  suggestion  which  applies  equally  well,  no  matter 
what  instrument  is  used  or  what  the  observation  may  be,  is : 

Before  going  into  the  field  the  observer  should  have  clearly 
and  definitely  in  mind  exactly  what  things  he  is  to  do,  and 
exactly  how  and  when  and  in  what  order  he  is  to  do  them.  This 
is  important. 


CHAPTER  VII 


OBSERVATIONS  FOR  LATITUDE 

40.  Latitude  by  a  Circumpolar  Star  at  Culmination.  This 
observation  consists  in  measuring  the  altitude  of  a  circumpolar 
star  at  upper  or  lower  culmination,  when  its  altitude  is  a  maximum 
or  a  minimum,  and  from  this  measured  altitude  and  known 
data  computing  the  altitude  of  the  north  celestial  pole,  which  is 
the  latitude  of  the  place.  Any  circumpolar  star  may  be  used, 
but  Polaris  is  the  best,  because  it  is  the  brightest. 

Referring  to  Fig.  16,  let  the  circle  represent  the  meridian  of 
the  observer,  who  is  at  0.  Let  HH'  and  EE'  represent  the 
projections  of  the  horizon  and  celestial  equator,  respectively, 

upon  the  plane  of  the 
meridian.  Let  Z  repre- 
sent the  zenith  and  P  the 
north  celestial  pole. 

By  definition,  the  arc 
EZ  (the  angular  distance 
of  the  observer  from  the 
equator)  is  the  observer's 
latitude.  It  is  obvious 
that  the  arc  H'P  is  equal 
to  the  arc  EZ,  and  that 
therefore  the  observer's 
latitude  may  be  defined 
as  the  declination  of  his 
FIG.  16.  zenith  (EZ}  or  as  the 

altitude,  with   respect   to 
his     horizon,     of     the    celestial    pole     (H'P). 

Let  S  and  S'  be  the  two  positions  of  a  circumpolar  star  when 
it  is  on  the  meridian  of  the  observer,  at  upper  and  lower  culmina- 
tion, respectively.  Considering  first  the  case  of  upper  culmina- 
tion, the  declination  of  the  star  is  ESt  and  its  altitude  is  H'S. 
Also: 

H'P  =  H'S  -  PS 

=  H'S  -  (90°  -  ES) 

or  0  =  h  -  (90°  -  5) (28) 

48 


LATITUDE   BY   A    CIRCUMPOLAR   STAR  49 

For  the  case  of  lower  culmination,  when  the  star  is  at  S't  the 
declination  is  E'S',  the  altitude  is  H'S'y  and: 

H'P  =  H'S'  +  PS' 
or  4>  =  h  +  (90°  -  6) (29) 

If  Polaris  is  the  star  used  it  is  not  strictly  necessary  that  the 
exact  time  of  culmination  be  known,  for  the  altitude  of  Polaris 
changes  but  very  sligktly  for  several  minutes  before  and  after 
culmination.  The  time  of  culmination  may  be  taken  from 
Table  V  (in  which  case  the  declination  may  be  taken  from 
Table  VI),  or  it  may  be  computed  more  exactly  by  the  following 
method.  This  method  applies  equally  well  to  any  circumpolar 
star. 

At  the  instant  of  upper  culmination  the  hour  angle,  t,  of  any 
star  is  equal  to  zero,  and  at  the  instant  of  lower  culmination  it 
is  equal  to  12  hours.  The  right  ascension  of  the  star  for  any 
desired  date  may  be  found  in  the  "American  Ephemeris  and 
Nautical  Almanac."  We  may  therefore  compute  the  sidereal 
time  of  culmination  by  the  following  equation: 

Sid.  T  =  R  A  +  t (24) 

This  sidereal  time  may  be  changed  to  standard  time  by  the 
methods  of  Arts.  31  and  27,  pages  34  and  31;  thus  giving  the 
standard  or  watch  time  of  culmination. 

The  longitude  of  the  place  may  be  obtained  with  sufficient 
accuracy  for  this  or  any  similar  computation  by  scaling  from 
one  of  the  government's  topographical  sheets  or  any  other  re- 
liable map,  even  if  drawn  to  a  very  small  scale.  An  error  of 
half  a  degree  in  longitude  represents  only  about  two  minutes' 
error  in  computed  time  of  culmination,  and  the  altitudes  of  any 
of  the  close  circumpolar  stars  change  but  slightly  for  several 
minutes  before  and  after  culmination. 

If  no  tables  or  means  of  computation  are  at  hand  the  approxi- 
mate time  of  upper  or  lower  culmination  of  Polaris  may  be 
estimated  by  the  relative  positions  of  Polaris  and  5  Cassiopeise. 
See  Art.  33,  page  38,  and  Fig.  12.  Beginning  some  little  time 
before  culmination,  the  motion  of  Polaris  may  be  followed  by 
the  tangent  screw  of  the  vertical  motion  of  the  transit,  bisecting 
the  star  with  the  horizontal  cross-hair,  until  it  has  reached  its 
highest  or  lowest  position  and  appears  to  have  only  a  horizontal 
motion.  Its  altitude  should  then  be  read. 

It  is,  of  course,  unnecessary  that  the  instrument  be  centered 
over  any  definite  station  during  this  observation,  as  a  difference 


50  OBSERVATIONS   FOR   LATITUDE 

of  a  minute  in  latitude  corresponds  to  about  6080  feet  on  the 
ground.  Either  an  engineer's  transit  or  a  sextant  and  artificial 
horizon  may  be  used  in  this  observation  if  the  time  of  culmina- 
tion has  been  computed. 

Outline  of  Observation: 

Computations  Preceding  Field  Work: 

Compute  time  of  U.  C.  or  of  L.  C.  of  star. 

From  Table  V  (for  Polaris  only),  or,  more  accurately, 

Sid.  T  =  R  A  +  t (24) 

R  A  from  "  Nautical  Almanac." 

t  =  Oh  for  U.  C.,  t  =  12"  for  L.  C. 
Change  Sid.  T  to  Std.  T. 
Field  Work: 

(A)  Using  a  sextant: 

Make  several  measurements  of  double  altitude  of 

star  within  three  minutes  of  time  of  culmination 
Determine  index  error  of  sextant. 

(B)  Using  a  transit  with  vertical  arc: 

Beginning  several  minutes  before  computed  time 
of  culmination,  follow  star  with  tangent  screw  to 
limit  of  star's  vertical  motion,  bisecting  it  with 
horizontal  cross-hair. 

Read  the  vertical  arc,  and  determine  index  error. 

(C)  Using  a  transit  with  vertical  circle: 

With  telescope  direct,  read  altitude  of  star  two  or 

three  minutes  before  culmination. 
Reverse    instrument   quickly  and  read   altitude   of 

star  with  telescope  inverted. 
Computations  Following  Field  Work: 

(A)  Apply  index  correction  to  measured  angle. 
Divide  result  by  two  for  apparent  altitude. 
Subtract  refraction  correction  from  apparent  altitude, 

thus  obtaining  true  altitude,  h. 

(B)  Correct  vertical  arc  reading  for  index  error  and  refrac- 

tion, thus  obtaining  true  altitude,  h. 

(C)  Subtract  refraction  correction  from  mean  of  two  read- 

ings, thus  obtaining  true  altitude,  h. 
Having  true  altitude,  h,  apply  equation: 

0  =  h  =F  (90°  -  5) (28),  (29) 


LATITUDE  BY  ALTITUDE  OF  A  SOUTHERN  STAR  51 


Use  —  sign  for  U.  C. 
Use  +  sign  for  L.  C. 
Obtain  d  from  Table  VI  (for  Polaris  only), 

or,  more  accurately  from  the  "Nautical 

Almanac." 

An  example  of  the  computations  and  field-notes  of  this  ob- 
servation is  given  on  pages  106  and  107. 

41.  Latitude  by  Meridian  Altitude  of  a  Southern  Star.    This 
observation  consists  in  measuring  the  altitude  of  a  southern 
star  when  it  is  on  the  ob- 
server's meridian — i.e.,  at 
upper  transit — and   from 
this     measured     altitude 
and  known  data  comput- 
ing the  declination  of  the 
observer's  zenith,  which  is 
his  latitude. 

Referring  to  Fig.  17,  a 
southern  star  may  cross 
the  meridian  between  the 
equator  and  the  zenith,  as 
at  S,  or  below  the  equator, 
as  at  S'.  The  lower 
transit  of  a  southern  star,  17* 

assuming  the  observer  to 
be    in     the     northern      hemisphere,      is     always     invisible. 

Considering  the  first  case,  the  declination,  ESt  is  positive;  and 

EZ  =  90°  -  (HS  -  ES), 
or  <f>  =  90°  -  (h  -  6) (30) 

If  the  star  is  at  S'  the  declination,  ES',  is  negative,  and  the 
equation  may  be  written: 

EZ  =  90°  -  HS'  -  (-  ESf) 

That  is,  if  6  is  always  substituted  with  its  proper  algebraic 
sign  the  equation  derived  above  is  general,  viz.: 

<j>  =  90°  -  (h  -  6) (30) 

The  sidereal  time  at  which  the  star  will  cross  the  meridian 
may  be  computed  from  the  equation: 

Sid.  T  =  R  A  +  t (24) 


52  OBSERVATIONS   FOR   LATITUDE 

the  right  ascension  of  the  star  for  the  proper  date  being  taken 
from  the  " American  Ephemeris  and  Nautical  Almanac,"  and 
the  hour  angle,  t,  for  the  instant  being  equal  to  0  hours.  The 
sidereal  time  thus  computed  may  be  changed  to  standard  time 
by  the  methods  of  Arts.  31  and  27,  pages  34  and  31. 

One  objection  to  this  observation  is  the  difficulty  in  identifying 
the  southern  stars.     If  the  direction  of  the  meridian  and  the 
approximate  latitude  are  known,  the  methods  of  identification 
of  Art.  48,  page  68,  may  be  used. 
Outline  of  Observation: 
Computations  Preceding  Field  Work: 
Compute  time  of  transit  of  star. 

Sid.  T  =  R  A  +  t 7  -r—      (24) 

R  A  from  "  Nautical  Almanac." 
t  =  0  hours. 

Change  Sid.  T  to  Std.  T. 
Field  Work: 

(A)  Using  a  sextant: 

Make    several  measurements  of  double  altitude  of 
star  within  two  or  three  minutes  of  time  of  transit. 
Determine  index  error  of  sextant. 

(B)  Using  a  transit  with  vertical  arc : 

Beginning  several  minutes  before  time  of  transit, 
follow  star  with  tangent  screw  to  limit  of  star's 
vertical  motion,  bisecting  it  with  horizontal  cross- 
hair. 

Read  the  vertical  arc  and  determine  index  error. 

(C)  Using  a  transit  with  vertical  circle: 

With  telescope  direct,  read  altitude  of  star  two  or 

three  minutes  before  transit. 
Reverse  instrument  quickly  and  read  altitude  of 

star  with  telescope  inverted. 
Computations  Following  Field  Work: 

(A)  Apply  index  correction  to  measured  angle. 
Divide  result  by  two  for  apparent  altitude. 
Subtract  refraction  correction  from  apparent  altitude, 

thus  obtaining  true  altitude,  h. 

(B)  Correct  vertical  arc  reading  for  index  error  and  refrac- 

tion, thus  obtaining  true  altitude,  h. 

(C)  Subtract  refraction  correction  from  mean  of  two  read- 

ings, thus  obtaining  true  altitude,  h. 


LATITUDE  BY  ALTITUDE  OF  THE  SUN       53 

Having  true  altitude,  h,  apply  equation: 

0  =  90°  -  (h  -  5) (30) 

Obtain  8  from  "  Nautical  Almanac,"  and 
substitute  it  with  proper  algebraic  sign. 

An  example  of  the  computations  and  field-notes  of  this  obser- 
vation is  given  on  pages  108  and  109. 

42.  Latitude  by  Meridian  Altitude  of  the  Sun.  The  methods 
and  equations  of  the  preceding  article  may  be  applied  to  the 
sun  as  well  as  to  a  southern  star.  The  altitude  of  one  edge,  or 
limb,  of  the  sun  instead  of  the  altitude  of  the  center  is  usually 
measured.  (See  Arts.  36  and  38,  pages  41  and  43.) 

The  watch  (standard)  time  of  transit  of  the  sun's  center  may 
be  obtained  by  changing  0  hours  local  apparent  time  to  standard 
time  by  the  method  of  Art.  29,  page  32.  If  the  direction  of  the 
meridian  is  known,  the  meridian  altitude  may  be  obtained  by 
use  of  a  transit  without  computation  of  the  time  of  transit  of 
the  sun,  by  setting  the  instrument  so  that  the  telescope  revolves 
in  the  plane  of  the  meridian  and  reading  the  altitude  of  one 
limb  of  the  sun  when  its  center  crosses  the  vertical  cross-hair. 
To  make  the  telescope  revolve  in  the  plane  of  the  meridian,  set 
up  the  transit  over  one  of  two  stakes  which  mark  the  direction 
of  the  meridian  and  sight  on  the  other.  The  line  of  sight  will 
now  revolve  (about  the  horizontal  axis  of  the  telescope)  in  the 
plane  of  the  meridian.  In  order  to  eliminate  the  effect  of  in- 
strumental errors  so  far  as  possible,  it  is  well  to  set  over  the 
farther  north  of  the  two  stakes  and  sight  on  the  south  stake. 

The  declination  of  the  sun  at  the  instant  at  which  the  obser- 
vation is  to  be  made  (local  apparent  noon)  may  be  obtained,  by 
interpolation,  from  the  "  Nautical  Almanac"  or  from  one  of 
the  instrument-makers'  reprints  from  the  "  Nautical  Almanac." 
The  longitude,  for  use  in  making  the  interpolation,  may  be 
obtained  by  scaling  from  some  reliable  map,  such  as  one  of  the 
government's  topographical  sheets.  This  value  of  the  longitude 
should  also  be  sufficiently  accurate  for  use  in  the  conversion  of 
time  mentioned  above. 
Outline  of  Observation: 
Computations  Preceding  Field  Work: 

Compute  the  standard  time  of  transit  of  the  sun's  center. 

Change  Ohours,  local  apparent  time,  to  standard  time/ 

Obtain  longitude  by  scaling  from  a  map, 

other  data  from  "  Nautical  Almanac." 


54  OBSERVATIONS   FOR   LATITUDE 

Field  Work: 

(A)  Using  a  sextant: 

Measure  double  altitude  of  lower  limb  of  sun  at  com- 
puted time  of  transit.  (See  Art.  39,  page  43,  for 
method  of  using  sextant.)  Determine  index  error. 

(B)  Using  a  transit: 

Follow  the  sun  with  tangent  screw  of  vertical  mo- 
tion as  long  as  it  rises,  keeping  the  horizontal 
cross-hair  tangent  to  the  sun's  lower  limb.  (See 
Art.  39,  page  43,  for  suggestions  in  regard  to 
sighting  on  the  sun.) 

Take  the  reading  of  the  vertical  arc  (or  circle)  cor- 
responding to  the  greatest  altitude  of  the  sun. 
(Should  occur  practically  at  computed  time  of 
transit.) 

Determine  index  error. 
Computations  Following  Field  Work: 

(A)  Apply  index  correction  to  measured  angle. 

Divide  result  by  two  for  apparent  altitude  of  lower  limb. 
.      Apply  refraction,  semi-diameter,  and  parallax  correc- 
tions, thus  obtaining  true  altitude,  h. 

(B)  Correct  vertical  arc  (or  circle)  reading  for  index  error, 

refraction,  semi-diameter,  and  parallax,  thus  obtain- 
ing true  altitude,  h. 

Having    true    altitude,  ft,  of   the   sun's  center,  apply 
equation : 

0  =  90°  -  (h  -  5)  .      .      .      '.     .     '.      (30) 

Obtain  6  for  the  time  of  observation 
from  the  "  Nautical  Almanac"  (using 
longitude  scaled  from  a  map  for  inter- 
polation), and  substitute  it  with  proper 
algebraic  sign. 

An  example  of  the  computations  and  field-notes  of  this  obser- 
vation is  given  on  pages  110  and  111. 


CHAPTER  VIII 

OBSERVATIONS  FOR  AZIMUTH 

For  the  engineer  in  general  practice  the  determination  of  true 
azimuth  is  probably  the  most  important  part  of  the  work  of 
field  astronomy. 

In  the  discussion  of  all  observations  for  azimuth  it  will  be 
assumed  that  the  transit  is  carefully  set  up  and  centered  over  a 
point  which  marks  one  end  of  a  line  whose  azimuth  is  desired. 
This  position  of  the  instrument  will  be  called  simply  the  "station." 
In  some  observations  it  will  be  most  convenient  to  line  in 
another  "point" — a  stake  and  tack — in  the  direction  of  a  star 
which  has  been  sighted,  and  whose  azimuth  at  the  instant  of 
sighting  can  be  computed;  in  other  observations  it  will  be  best 
to  measure  the  angle  from  the  star  or  the  sun  to  a  signal.  In 
any  case,  the  true  azimuth  of  a  line  on  the  ground,  one  end  of 
which  is  the  station  and  the  other  a  stake  or  signal,  is  obtained. 
From  this  reference  line  we  may  then  determine  the  azimuth  of 
any  other  line  which  passes  through  the  station,  or  we  may 
determine  the  direction  of  the  meridian  through  the  station. 
It  is  customary  to  call  the  angle  between  the  true  north  and  the 
direction  of  a  circumpolar  star  the  azimuth  of  the  star,  without 
regard  to  whether  the  star  is  east  or  west  of  the  true  north. 
This  azimuth  is  the  angle  Z  of  the  astronomical  triangle. 

If  a  stake  is  to  be  lined  in  at  night  it  may  be  conveniently 
done  in  the  following  manner: 

First  line  in  a  lantern.  Then  hold  in  front  of  the  lantern  an 
oiled  paper  screen,  and  in  front  of  the  screen  a  stake.  (A  suit- 
able screen  may  be  made  by  tacking  some  heavy  paper  to  four 
sticks  nailed  together  to  form  a  rectangular  frame,  and  pouring 
some  kerosene  or  other  oil  over  the  paper;  or  a  handkerchief 
will  do  fairly  well  in  place  of  a  screen.)  The  stake  can  then  be 
seen,  black  against  the  bright  screen,  and  lined  in  in  the  ordinary 
manner.  The  cross-hairs  of  the  transit  can  often  be  seen  against 
the  screen,  thus  avoiding  the  necessity  of  otherwise  illuminating 
them.  The  stake  should  be  driven  where  a  pencil  held  on  the 
top  can  be  seen,  to  be  lined  in  for  the  exact  point.  The  screen 
is  not  necessary,  but  it  furnishes  a  larger,  more  uniformly  lighted 
area  against  which  to  see  the  stake  than  does  the  lantern  alone. 

55 


56  OBSERVATIONS   FOR   AZIMUTH 

A  suitable  signal,  or  "azimuth  mark,"  from  which  to  measure 
an  angle  to  a  star,  may  be  made  from  a  wooden  box  large  enough 
to  hold  a  lantern.  A  hole  should  be  bored  in  the  front  of  the  box 
at  the  height  of  the  blaze  of  the  lantern,  the  size  of  the  hole 
depending  on  the  distance  from  the  station  at  which  the  box 
is  to  be  set,  and  on  the  strength  of  the  light  behind  it.  These 
factors  should  be  so  adjusted  that  the  appearance  at  night  will 
be  that  of  a  point  of  light  not  unlike  a  star.  If  an  ordinary 
kerosene  lantern  is  used,  a  hole  half  an  inch  in  diameter  in  a  box 
a  quarter  of  a  mile  or  so  away  will  give  a  suitable  mark  to  sight 
on.  The  distance  from  the  station  should  be  great  enough  so 
that  the  focus  of  the  telescope  will  not  have  to  be  changed 
when  sighting  first  at  the  signal  and  then  at  a  star.  A  vertical 
line  through  the  hole  which  can  be  used  as  a  sight  from  the 
station  in  the  daytime  should  be  painted  on  the  box.  The  box 
should  be  covered  to  prevent  the  lantern  being  blown  out;  and 
it  should  be  nailed  to  a  tree  or  to  stakes  driven  firmly  in  the 
ground  approximately  north  of  the  station,  and  where  an  un- 
obstructed view  of  it  may  be  had  from  the  instrument. 

It  is  a  good  plan  to  line  in  a  stake  between  the  station  and 
the  mark  and  near  the  latter,  so  that  its  direction  will  not  be 
lost  if  the  box  is  destroyed. 

43.  Azimuth  by  a  Circumpolar  Star  at  Elongation.  This 
method  is  probably  under  ordinary  conditions  the  most  reliable 
means  of  determining  true  azimuth. 

The  observation  consists  in  measuring  the  angle  from  a  circum- 
polar  star  at  elongation  (eastern  or  western)  to  an  azimuth  mark, 
and  from  the  measured  angle  and  the  computed  azimuth  of  the 
star  at  the  instant  of  elongation  computing  the  azimuth  of  the 
mark  from  the  station.  If  desired,  instead  of  measuring  the 
angle  to  an  azimuth  mark,  a  stake  and  tack  may  be  lined  in  in 
the  direction  of  the  star.  Any  circumpolar  star  may  be  used, 
but  Polaris  is  the  best.  If  Polaris  is  used  the  time  of  elongation 
may  be  taken  from  Table  V;  or  it  may  be  more  accurately  com- 
puted by  the  following  method,  which  applies  to  any  circumpolar 
star. 

The  sidereal  time  of  elongation  may  be  obtained  from  the 
equation: 

Sid.  T  =  R  A  +  t (24) 

The  right  ascension  for  the  proper  date  may  be  taken  from 
the  "Nautical  Almanac,"  and  the  value  of  t  found  by  a  solution 


AZIMUTH   BY   A    CIRCUMPOLAR   STAR  57 

of  the  astronomical  triangle.  See  Art.  12,  Solution  (2),  page  14, 
and  Art.  9,  pages  9  to  12.  The  sidereal  time  thus  obtained  may 
be  changed  to  standard  time  by  the  methods  of  Arts.  31  and  27, 
pages  34  and  31. 

If  no  tables  or  means  of  computation  are  at  hand,  the  approxi- 
mate time  of  elongation  of  Polaris  may  be  estimated  by  the 
relative  positions  of  Polaris  and  6  Cassiopeise  (see  Art.  33,  page  38, 
and  Fig.  13)  and  the  motion  of  Polaris  followed  with  the  tangent 
screw  of  one  of  the  horizontal  motions  of  the  transit  (bisecting 
the  star  with  the  vertical  cross-hair)  until  it  ceases  to  move  east 
or  west  as  the  case  may  be,  and  appears  to  be  moving  vertically. 

The  azimuth  of  Polaris  when  at  elongation  may  be  taken  from 
Table  VII,  or  it  may  be  more  accurately  computed  by  Formula 
(12),  Art.  12,  page  15;  the  declination  being  taken  from  the 
"  Nautical  Almanac,"  for  the  proper  date  and  the  latitude 
being  known  from  a  previous  observation  or  from  an  accurate 
map. 

Outline  of  Observation: 

Computations  Preceding  Field  Work: 

Compute  time  of  elongation  of  star — eastern  or  western. 
From  Table  V  (for  Polaris  only),  or  more  accurately, 

Sid.  T  =  R  A  +  t (24) 

R  A  from  "  Nautical  Almanac." 
t  =  P  (western  elongation)  or 
t  ==  24 h  —  P  (eastern  elongation). 

cos  P  =  ^^  (13) 

tan  5 

4>  from  a  previous  observation  or  from  a 

map  . 

6  from  "Nautical  Almanac." 
Change  Sid.  T  to  Std.  T. 
Field  Work: 

Sight  on  azimuth  mark  with  plates  set  at  zero. 
Turn  to  star  with  upper  motion  of  transit  and  follow  star 
with  upper   tangent  screw,  bisecting  star  with  vertical 
cross-hair. 

V.      Three  or  four  minutes  before  time  of  elongation,  when  star 
appears  to  move  vertically,   read   the  horizontal  angle 
between  mark  and  star. 
Double  the  angle. 


58  OBSERVATIONS   FOR   AZIMUTH 

Field  Work   (Continued): 

Reverse  the    instrument    quickly    and    double    the    angle 

with  telescope  inverted. 
Computations  Following  Field  Work: 

Compute  azimuth  of  star  at  elongation. 

From  Table  VII  (for  Polaris  only),  or  more  accurately, 


COS0 

5  from  "  Nautical  Almanac." 
<f>  from  a  previous  observation  or  from  a  map. 
From  computed  azimuth  of  star  and  mean  of  measured  angles 
between  star  and  azimuth  mark,  compute  the  azimuth  of 
the  mark. 
Or,  if  desired,  the  field  work  may  be  done  as  follows: 

Field  Work: 

Sight  on  star  and  follow  it  with  tangent  screw  of  either 

horizontal  motion  of  transit  until  two  or  three  minutes 

before  elongation,  when  star  appears  to  move  vertically. 

Plunge   telescope   down   and   line    in   a   stake   and    tack 

several  hundred  feet  away. 
Reverse  instrument  quickly,  sight  on   star   with  telescope 

inverted,   and  line  in   another   point  beside  the  first. 
The  mean  of  these  two  points  should  be  in  the  direction 

of  the  star  at  elongation. 
Computations  as  outlined  above. 

An  example  of  the  computations  and  field-notes  of  this  obser- 
vation is  given  on  pages  112  and  113. 

44.  Azimuth  by  Polaris  Near  Elongation.  If  the  observation 
described  in  the  preceding  article  is  made  on  Polaris  within  thirty 
minutes  of  elongation,  the  azimuth  of  the  star  at  the  instant  of 
each  sight  may  be  obtained  from  the  computed  azimuth  at 
elongation  by  the  following  formula: 

C  =  3600  X  112.5  X  sin  V  X  tan  Ze  X  T2      .     (31)* 

where  T  is  the  interval  in  (sidereal)  minutes  between  the  instant 
of  elongation  and  the  instant  of  sighting. 

*  The  demonstration  of  this  formula  may  be  found  in  more  complete  works 
on  field  astronomy;  for  instance,  Doolittle's  "  Practical  Astronomy." 


AZIMUTH   BY   POLARIS  NEAR   ELONGATION  59 

Ze  is  the  azimuth  of  Polaris  at  elongation,  to  be  computed. 
C  is  the  correction  in  seconds  of  arc,  to  be  subtracted 
from  azimuth  at  elongation. 

Table  VIII  gives  the  values  of  "  C  "  of  the  above  formula  for 
each  minute  (of  the  interval  between  the  instant  of  elongation 
and  the  instant  of  sighting)  up  to  thirty  minutes,  and  for  values 
of  Ze  from  1°  10'  to  2°  10'.  The  corrections  are  also  given  in 
Table  Va  near  the  back  of  the  "  Nautical  Almanac." 

This  observation  is  a  convenient  one  to  use  when  the  time 
of  elongation  of  Polaris  comes  a  few  minutes  before  (or,  in  the 
morning,  after)  it  is  dark  enough  for  the  star  to  be  seen  through 
the  telescope.  The  work  of  this  observation  is  practically  the 
same  as  that  outlined  in  the  preceding  article,  with  the  addition 
that  the  time  of  each  pointing  of  the  telescope  at  the  star  must 
be  taken — at  least  as  accurately  as  to  the  nearest  minute — in 
order  to  obtain  the  interval  "  T"  of  Formula  (31)  between  time 
of  elongation  and  time  of  sighting.  Though  T  should  theoreti- 
cally be  a  number  of  sidereal  minutes,  no  appreciable  error  will 
result  from  using  T  as  the  number  of  solar  minutes,  as  obtained 
directly  from  the  computed  time  of  culmination  and  the  watch 
readings. 

Outline  of  Observation: 
Computations  Preceding  Field  Work: 

Exactly  as  outlined  in  Art.  43,  page  57. 

Field  Work: 

As  near  time  of  elongation  as  practicable,  sight  on  azimuth 
mark  with  plates  set  at  zero. 

Measure  the  angle  from  the  mark  to  the  star  four  times  by 
the  method  of  repetition  (the  final  reading  on  the  plates 
should  be  four  times  the  value  of  the  angle),  twice  with 
the  telescope  direct  and  twice  with  it  inverted.  Read 
the  plates  at  the  first  and  fourth  settings  on  the  star. 

Take  the  watch  reading  (standard  time)  to  nearest  half- 
minute  at  each  setting  on  the  star. 

Computations  Following  Field  Work: 

Compute    azimuth   of    star    at   elongation  as  outlined   in 

Art.  43,  page  58. 

Using    computed    time  of  elongation  and  mean  of  watch 
readings,  compute  "  T." 


60  OBSERVATIONS    FOR   AZIMUTH 

Computations  Following  Field  Work: 

Using  the  computed  azimuth  at  elongation  and  the  correc- 
tion "C"  from  Table  VIII,  compute  the  azimuth  of 
the  star  corresponding  to  the  mean  of  the  watch  read- 
ings. (Subtract  "C"  from  azimuth  at  elongation.) 
Using  this  computed  mean  azimuth  and  the  mean  angle 
from  the  mark  to  the  star,  compute  the  azimuth  of  the 
mark. 

An  example  of  the  computations  and  field-notes  of  this  obser- 
vation is  given  on  pages  114  and  115. 

45.  Azimuth  by  a  Circumpolar  Star  at  any  Hour  Angle.  This 
observation  is  one  of  the  most  precise  methods  for  the  deter- 
mination of  azimuth.  The  difficulty  in  its  use  in  practice  under 
ordinary  conditions  is  in  obtaining  the  standard  time  with 
sufficient  accuracy.  Its  advantage  over  an  observation  at 
elongation  is  that  the  number  of  observations  may  be  increased 
indefinitely,  thereby  securing  greater  precision. 

The  observation  consists  in  measuring  a  series  of  angles  be- 
tween an  azimuth  mark  and  a  circumpolar  star,  taking  the  time 
of  each  pointing  of  the  telescope  at  the  star;  and  from  the  mean 
of  the  measured  angles  and  the  azimuth  of  the  star  computed  for 
the  mean  of  the  recorded  times,  computing  the  azimuth  of  the 
mark.  In  work  done  with  great  precision  with  large  instruments 
several  corrections  need  to  be  introduced  which  are  too  small 
to  be  considered  in  work  done  with  an  engineer's  transit.  Any 
of  the  close  circumpolar  stars  may  be  used  for  the  observation. 
Polaris  is  the  best. 

The  mean  of  the  recorded  times  of  setting,  read  in  standard 
time  as  accurately  as  possible, — correct  within  a  very  few  seconds 
— may  be  changed  to  sidereal  time  by  the  methods  of  Arts.  26 
and  30,  pages  31  and  33,  and  Equation  (24),  from  Art.  21, 
page  23,  applied: 

Sid.  T  =  R  A  +  t (24) 

The  right  ascension  of  the  star  having  been  obtained  from 
the  "Nautical  Almanac,"  this  formula  may  be  used  to  compute 
the  hour  angle  of  the  star  at  the  mean  of  the  recorded  times  of 
setting. 

The  latitude  being  known,  the  declination  of  the  star  having 
been  obtained  from  the  "  Nautical  Almanac,"  and  the  hour 
angle  having  been  computed,  the  astronomical  triangle  may  be 


AZIMUTH   BY   A    STAR   AT   ANY   HOUR   ANGLE          61 

solved  for  the  azimuth  of  the  star  by  the  following  formula 
(from  Appendix  A)  : 

sin  t 

tan  Z  =  -  -     .     .      (10) 

tan  5  .  cos  <f>  —  sin  </>  .  cos  t 

If  the  latitude  of  the  station  is  not  known  the  altitude  of  the 
star  may  be  read  at  the  beginning  and  at  the  end  of  each  set  of 
readings  of  horizontal  angles  —  one  set  being  taken  with  telescope 
direct  and  one  with  telescope  inverted  —  and  the  mean  of  these 
four  altitudes,  corrected  for  refraction,  may  be  used  in  solving 
the  astronomical  triangle  for  azimuth  by  the  following  formula 
(from  Appendix  A)  : 

sin  t  .  cos  5 

sin  Z  =  -  ....      (11) 

cosh 

Outline  of  Observation: 
Computations  Preceding  Field  Work: 

None. 
Field  Work: 

By  method  of  repetition  read  two  sets  of  three  or  four 
angles  each  from  azimuth  mark  to  star,  one  set  with  tel- 
escope direct,  and  one  with  telescope  inverted;  reading 
the  watch  (Std.  T)  at  the  instant  of  each  pointing  at  the 
star.  Only  three  readings  from  the  transit  plates  need 
be  made:  the  value  of  the  first  angle  and  the  reading  at 
the  end  of  each  set. 

If  it  is  desired  to  solve  the  astronomical  triangle  by  the  sec- 
ond method  suggested  above  (latitude  unknown),  the  alti- 
tude of  the  star  should  be  read  at  the  beginning  and 
end  of  each  set. 
Computations  Following  Field  Work: 

Compute  hour  angle  of  star  at  mean  of  observed  times. 

t  =  Sid.T-RA       ........      (24) 

Sid.  T  from  mean  of  watch  readings  (Std.  T), 

changed  to  Sid.  T. 
R  A  from  "  Nautical  Almanac." 
Compute  azimuth  of  star  at  mean  of  observed  times. 


tan  Z  =  --        -  (10) 

tan  5  .  cos  0  —  sin  <£  .  cos  t 


62 


OBSERVATIONS   FOR   AZIMUTH 


t  from  computation  above. 
5  from  "  Nautical  Almanac." 
<£  from  a  former  observation  or  from  an  ac- 
curate map. 

Compute  azimuth  of  mark,  using  mean  azimuth  of  star 
from  above  computation  and  mean  of  observed  hori- 
zontal angles. 

An  example  of  the  computations  and  field-notes  of  this  obser- 
vation is  given  on  pages  116  and  117. 

46.  Azimuth  by  an  Altitude  of  the  Sun  or  of  a  Star.  This 
observation  consists  in  taking  a  series  of  sights  on  the  sun  at 
each  of  which  the  time,  the  sun's  altitude,  and  the  horizontal 
angle  between  the  sun  and  a  reference  mark  are  read.  From 
the  mean  of  the  observed  times,  the  mean  of  the  altitudes  and 
data  either  known  or  given  in  the  "  Nautical  Almanac,"  the 
mean  azimuth  of  the  sun  may  be  computed.  This  mean  azimuth 


FIG.  18. 

combined  with  the  mean  of  the  measured  angles  between  the 
sun  and  the  reference  mark  will  give  the  azimuth  of  the  mark. 

The  setting  of  the  cross-hairs  tangent  to  the  sun's  disc  may 
be  conveniently  done  in  the  following  manner: 

The  transit  having  been  centered  over  the  station,  sighted 
at  the  reference  mark  with  the  plates  set  at  zero,  the  upper  clamp 
loosened,  and  the  telescope  turned  toward  the  sun:  If  the 
observation  is  being  made  in  the  forenoon  the  sun's  disc  should 
first  be  brought  into  the  position  shown  in  Fig.  18  and  the 
horizontal  and  vertical  motions  clamped.  The  arrow  shows  the 
direction  of  the  sun's  apparent  motion.  The  vertical  cross- 
hair should  be  kept  tangent  to  the  sun's  disc  by  use  of  the  tangent 


AZIMUTH   BY   AN   ALTITUDE   OF   THE   SUN  63 

screw  of  the  upper  horizontal  motion  of  the  transit  until  the 
upward  motion  of  the  sun  has  brought  the  disc  tangent  to  the 
horizontal  cross-hair.  At  this  instant  the  time  should  be  noted, 
and  the  horizontal  and  vertical  circles  read.  This  operation 
should  be  repeated  quickly,  and  then  the  same  number  of  settings 
should  be  made  with  the  sun  in  the  diagonally  opposite  quadrant, 
as  shown  in  Fig.  19. 

This  time  it  will  be  convenient  to  keep  the  horizontal  cross- 
hair tangent  with  the  tangent  screw  of  the  vertical  motion  of 
the  transit,  letting  the  horizontal  movement  of  the  sun  bring 
the  disc  tangent  to  the  vertical  cross-hair. 

It  is  evident  that  the  mean  of  the  four  altitudes  and  of  the 
four  horizontal  angles  will  not  have  to  be  corrected  for  semi- 
diameter  of  the  sun.  If  the  transit  has  a  full  vertical  circle  the 
telescope  should  be  inverted  between  the  two  sets;  if  not,  index 
correction  must  be  made.  The  instrument  should  be  very 
carefully  leveled  for  this  observation.  Care  should  be  taken 
not  to  use  one  of  the  stadia  wires  for  the  middle  cross-hair. 
One  of  the  schemes  for  protecting  the  eye  from  the  sun  which 
are  suggested  in  Art.  39,  page  44 — covering  the  eye-piece  with 
either  a  colored  glass  or  a  prismatic  eye-piece  which  has  a  colored 
glass  or  projecting  the  image  on  to  a  card — may  be  used. 

It  should  be  remembered  that  if  a  non-erecting  telescope  is 
used  the  direction  of  the  sun's  apparent  motion  will  be  reversed; 
and  that  if  a  prismatic  eye-piece  is  used  the  sun's  image  will  be 
turned  upside  down,  but  not  left  for  right.  However,  this  need 
not  complicate  matters,  since  it  makes  no  difference  in  which 
quadrants  the  sun's  image  is  placed  if  only  the  same  number  of 
settings  is  made  with  the  sun  in  each  of  two  diagonally  opposite 
quadrants.  If  the  observation  is  being  made  in  the  afternoon 
the  relative  positions  of  the  sun's  image  and  the  cross-hairs  a 
few  seconds  before  becoming  tangent  should  appear  through 
an  erecting  eye-piece,  as  shown  in  Fig.  20.  The  procedure  should 
be  obvious. 

The  approximate  longitude  being  known  (scaled  from  a  map), 
the  sun's  declination  for  the  instant  of  the  mean  of  the  watch 
readings  may  be  obtained  by  interpolation  from  the  "  Nautical 
Almanac."  The  mean  of  the  observed  altitudes,  corrected  for 
index  error  if  necessary  and  for  refraction  and  parallax,  gives 
the  altitude  of  the  sun's  center  at  the  mean  of  the  observed 
times.  These,  with  the  latitude  (from  a  previous  observation 
or  from  a  map),  furnish  data  for  the  solution  of  the  astronomical 


64 


OBSERVATIONS    FOR   AZIMUTH 


triangle  for  azimuth  by  one  of  the  formulas  of  Art.  12,  page  13. 
If  the  latitude  is  not  known,  the  hour  angle  of  the  sun's  center 
may  be  obtained  by  changing  the  mean  of  the  watch  readings 
(standard  time)  to  local  apparent  time,  and  used  instead  of  the 
latitude  in  the  solution  of  the  triangle  by  Equation  (11),  (from 
Appendix  A) .  Under  ordinary  field  conditions  this  latter  solution 
is  likely  to  give  less  accurate  results  than  the  use  of  the  latitude. 


FIG.  20. 


The  mean  azimuth  of  the  sun  thus  computed,  combined  with 
the  mean  of  the  readings  from  the  horizontal  circle  of  the  transit, 
will  give  the  azimuth  of  the  reference  mark. 

For  good  results  this  observation  should  not  be  made  within 
two  hours  of  noon  or  when  the  sun's  altitude  is  less  than  10°  or  15°. 

The  methods  of  this  observation  may  be  applied  equally  well 
to  an  observation  on  a  star,  the  star's  image  being  bisected  at 
each  setting  by  both  horizontal  and  vertical  cross-hairs.  The 
declination  of  a  star  changes  so  slowly  that  it  may  be  considered 
constant  for  the  day,  so  the  exact  time  of  each  setting  need  not 
be  taken.  Of  course,  no  parallax  correction  need  be  applied 
to  the  altitude  of  the  star. 

Outline  of  Observation: 
Computations  Preceding  Field  Work: 

None. 
Field  Work: 

Sight  at  reference  mark  with  plates  clamped  at  zero. 
Loosen  upper  clamp  and  turn  telescope  toward  sun. 
Make  two  settings  of  cross-hairs  tangent  to  right  and  lower 
limbs  of  sun's  disc  (if  in  A.M.);  recording  at  each  setting: 


AZIMUTH   BY   EQUAL   ALTITUDES   OF   A   STAR         65 

(1)  watch  reading,  (2)  horizontal  circle  reading,  (3)  vertical 

circle  reading. 
Repeat  the  settings  (with  telescope  inverted  if  instrument 

has  a  full  vertical  circle),  making  the  cross-hairs  tangent 

to  the  left  and  upper  limbs. 
If    instrument    has    only  a  vertical  arc    determine    index 

correction. 
Turn  back  to  reference  mark  and  check  back-sight. 

Computations  Following  Field  Work: 

Compute  azimuth  of  sun's  center  at  instant  of  mean  of  watch 
readings: 

Formula  from  Art.  12,  page  14,  preferably: 


cos  (k  +  0)  .  cos  (k  +  h) 
sin  k  .  cos  (k  -f-  5) 

<f>  from  a  previous  observation  or  from  a  map. 
h  from  mean  of  observed  altitudes,  corrected  for 
refraction  and  parallax,  and  for  index  error  if 
necessary. 

5  for  instant  of  mean  of  watch  readings  by  inter- 
polation from  "  Nautical  Almanac." 
k  =  J  [270°  -(</>+  h  +  5)]. 

From  computed  mean  azimuth  of  sun  and  mean  of  horizon- 
tal circle  readings,  compute  azimuth  of  reference  mark. 
It  should  be  remembered  that  Z  in  the  above  formula  is  not 
always  the  azimuth,  but  is  an  interior  angle  of  the  astro- 
nomical triangle. 

An  example  of  the  computations  and  field-notes  of  this  obser- 
vation is  given  on  pages  118  and  119. 

47.  Azimuth  by  Equal  Altitudes  of  a  Star.  This  observation 
consists  in  marking  by  a  stake  and  tack,  set  several  hundred 
feet  from  the  station,  the  direction  of  a  star  two  or  three  hours 
before  its  transit  (upper  or  lower),  the  altitude  of  the  star  being 
noted;  and  then  marking  by  another  stake  and  tack  the  direction 
of  the  same  star  at  the  instant  when  it  reaches  the  same  altitude 
on  the  opposite  side  of  the  meridian.  The  bisector  of  the  angle 
between  these  two  directions  is  the  true  meridian  through  the 
station. 

The  chief  advantages  of  this  observation  are  that  no  tables 
or  computations  and  no  information  as  to  standard  time  are 


66  OBSERVATIONS    FOR   AZIMUTH 

necessary;  and  a  high  degree  of  precision  is  attainable  by  a 
sufficient  number  of  repetitions.  It  is  inconvenient  in  that  the 
two  series  of  observations  must  be  made  at  an  interval  of  at 
least  four  to  six  hours. 

Instead  of  bisecting  the  star  with  the  horizontal  and  vertical 
cross-hairs  and  attempting  to  read  the  vertical  circle  during 
the  first  series  of  sights  (before  transit);  it  is  probably  better 
to  set  the  vertical  circle  at  some  exact  minute,  so  that  the 
horizontal  cross-hair  is  just  below  the  star  if  it  is  moving  down 
or  just  above  it  if  it  is  rising,  and  to  follow  the  movement  of  the 
star  with  the  tangent  screw  of  one  of  the  horizontal  motions 
(bisecting  the  star  with  the  vertical  wire);  so  that  when  the 
vertical  motion  of  the  star  has  brought  it  to  the  horizontal  cross- 
hair it  will  be  exactly  at  the  intersection  of  the  two.  If  the 
instrument  has  a  vertical  circle  the  first  series  of  observations 
should  be  made  with  the  telescope  direct  and  the  second  series 
with  it  inverted.  The  refraction  correction  need  not  be  con- 
sidered, since  its  effect  on  the  two  series  of  observations  is  exactly 
compensating. 

Any  star  may  be  used  which  is  at  a  convenient  altitude  two 
or  three  hours  before  the  time  of  its  transit,  and  which  will  come 
to  the  same  altitude  on  the  other  side  of  the  meridian  (four  to 
six  hours  later)  before  daylight  interferes  with  the  observation. 
One  of  the  stars  of  Cassiopeia  or  of  Ursa  Major  may  ordinarily 
be  used. 

Outline  of  Observation: 
Computations  Preceding  Field  Work: 

None. 
Field  Work: 

Set  the  vertical  circle  or  arc  to  read  some  exact  minute 

so  that  the  star  selected  is  approaching  the  horizontal 

cross-hair. 

Record  this  reading. 
Bisect  the  star  with    the   vertical    cross-hair    and   follow 

it    with    the   tangent    screw  of    one   of    the   horizontal 

motions  of  the  transit  until  it  is  at  the  intersection  of 

the  cross-hairs. 
Line  in  and   center  a  stake  three  or  four  hundred  feet 

from  the  station. 
Repeat  this  operation  two  or  three  times  at  intervals  of 

ten  or  fifteen  minutes,   marking   the  successive  stakes 

"A,"  "B,""C,"etc. 


AZIMUTH   BY   EQUAL   ALTITUDES   OF  A   STAR         67 

When  the  star  is  approaching  the  same  altitude  on  the 
other  side  of  the  meridian,  set  the  vertical  circle  or  arc 
to  read  the  last  altitude  used.  (If  the  transit  has  a 
vertical  circle,  have  the  telescope  inverted.) 

Bisect  the  star  with  the  vertical  cross-hair  and  follow  it  to 
the  intersection  of  the  cross-hairs  as  before. 

Set  and  center  a  stake  three  or  four  hundred  feet  from 
the  station,  marking  it  to  correspond  with  the  last 
stake  set. 

Using  the  altitudes  used  in  the  first  series  of  observations 
in  the  reverse  order  from  that  in  which  they  were  ob- 
tained, set  as  many  stakes  as  in  the  first  series,  marking 
them  ***  "C,"  "B,"  "A." 

Bisect  the  angle  "A-station-A,"  and  set  and  center  a  broad- 
topped  stake  three  or  four  hundred  feet  from  the  station. 

Bisect  each  of  the  other  angles  (B-station-B,  C-station-C, 
etc.)  and  set  points  on  the  broad-topped  stake  beside 
the  first. 

All  of  these  points  should  coincide,  and  a  line  through  the 
station   and    the    point    of    coincidence    should   be    the 
meridian.     If  they  do  not  coincide,   the  mean  of  their 
positions  should  be  used. 
Computations  Following  Field  Work: 

None. 

An  example  of  the  field-notes  of  this  observation  is  given  on 
pages  120  and  121. 


CHAPTER  IX 
OBSERVATIONS  FOR  TIME 

48.  Time  by  Transit  of  a  Star.  This  observation  consists  in 
noting  the  watch  time  of  transit  of  a  star  over  the  observer's 
meridian.  The  sidereal  time  of  transit  of  a  star  may  be  obtained 
from  its  right  ascension  (taken  from  the  "  Nautical  Almanac," 
and  substituted  in  the  equation:  Sid.  T  =  R  A  -f- 1)  and 
changed  to  standard  time.  The  difference  between  this  standard 
time  and  the  watch  reading  at  the  instant  of  transit  is  the  watch 
correction. 

Stars  to  be  used  for  the  determination  of  time  should  move 
rapidly,  and  should  therefore  be  those  which  are  as  near  as 
practicable  to  the  equator.  Without  a  prismatic  eye-piece — 
which  is  inconvenient  for  night  work — stars  at  altitudes  greater 
than  50°  or  55°  can  not  easily  be  seen  with  a  transit;  and  it  is 
not  well  to  try  to  use  stars  whose  meridian  altitudes  are  less 
than  15°  01  20°,  if  indeed  the  topography  (surrounding  mountains, 
etc.)  does  not  make  a  higher  minimum  necessary. 

Smaller  stars  than  those  of  the  fifth  magnitude  should  not 
be  selected  as  they  are  too  difficult  to  observe  with  the  ordinary 
transit  telescope 

Before  going  into  the  field  the  observer  should  prepare  a  list  of 
suitable  stars,  the  list  giving  the  name  of  the  star,  its  magnitude, 
the  local  mear  time  of  transit,  and  its  approximate  meridian 
altitude.  Standard  time  may  be  used  in  place  of  local  mean 
time  if  the  longitude  of  the  place  is  known  as  accurately  as  re- 
sults are  desired.  If,  for  instance,  it  is  desired  to  make  the 
observations  between  eight  and  nine  o'clock,  P.M.,  on  a  certain 
date,  these  two  hours  changed  into  sidereal  time  (at  least  the 
approximate  longitude  being  known)  will  give  the  limiting 
values  of  right  ascension.  This  comes  from  the  relation: 

Sid.T  =RA  +  t (24) 

t  being  equal  to  0  hours  at  the  time  of  transit. 

The  limiting  altitudes,  considered  with  the  latitude  of  the 
observer,  will  determine  the  limiting  values  of  declination. 

Having  determined  the  limiting  values  of  right  ascension, 
68 


TIME    BY   TRANSIT   OF   A   STAR  69 

declination,  and  magnitude,  the  stars  may  be  selected  from  the 
lists  given  in  the  "  Nautical  Almanac,"  and  times  of  transit 
and  approximate  meridian  altitudes  computed.  Refraction 
may  be  neglected  in  computing  the  approximate  altitudes. 

If  the  line  of  sight  is  now  put  in  the  plane  of  the  meridian, 
and  the  approximate  altitude  of  one  of  the  stars — the  first  on 
the  list  to  pass  the  meridian — set  oft7  on  the  vertical  arc,  the  star 
may  be  identified  because  it  will  cross  the  field  of  view  following 
very  nearly  along  the  horizontal  cross-hair.  If  the  watch  is 
keeping  approximately  local  mean  time,  or  if  it  is  keeping  ap- 
proximately standard  time  and  the  longitude  of  the  place  is 
known  so  that  approximate  local  mean  time  may  be  computed 
from  the  watch  reading,  it  will  assist  in  the  identification  and 
shorten  the  time  of  waiting. 

If  the  transit  has  a  vertical  circle,  half  the  observations  should 
be  made  with  the  telescope  direct  and  half  with  it  inverted,  the 
mean  of  the  determined  watch  corrections  being  accepted.  In 
any  case,  great  care  must  be  used  in  leveling  the  instrument, 
especially  in  a  direction  parallel  to  the  horizontal  axis.  A  striding 
level  may  be  used  to  good  advantage. 

Outline  of  Observation: 

Computations  Preceding  Field  Work: 

Prepare  a  list  of  four  or  six  stars  of  at  least  fifth  magnitude 
with  meridian  latitudes  between  10°  and  55°,  and  with 
convenient  times  of  transit. 

Meridian  altitude  of  a  southern  star  from: 

h  =  90°  -  <£  +  5 (32) 

Substitute  5  with  proper  algebraic  sign. 
Local  mean  time  of  transit  from: 

Sid.T  =RA+t .     (24) 

R  A  from  "  Nautical  Almanac." 
t  =  0  hours. 

Change  Sid.  T  to  LMT. 
Field  Work: 

Having  the  meridian  marked  by  two  stakes,  set  the  transit 
over  one  and  sight  on  the  other.  The  line  of  sight  should 
now  revolve  (about  the  horizontal  axis)  in  the  plane  of 
the  meridian. 

Set  off  on  the  vertical  arc  the  approximate  meridian 
altitude  of  the  first  star  on  the  list  (arranged  in  order 
of  transit). 


70  OBSERVATIONS    FOR   TIME 

Field  Work: 

Note  and  record  the  watch  reading  at  instant   of  transit 

of  star  across  vertical  cross-hair. 

Repeat  the  operation  for  the  other  stars  on  the  list,  if  pos- 
sible making  half  the  observations  with  telescope  direct 
and  half  with  it  inverted. 
Computations  Following  Field  Work: 

Determine    the    difference    between    the    computed    local 
mean  time  of  transit  and  the  watch  reading  at  the  instant 
of  transit  for  each  star  observed. 
Take  the  mean  of  the  computed  differences  for  the  watch 

correction  to  local  mean  time. 

An  example  of  the  computations  and  field  notes  of  this  obser- 
vation is  given  on  pages  122  and  123. 

49.  Time  by  Transit  of  the  Sun.  This  observation  consists  in 
noting  the  watch  times  of  transit  of  the  west  and  east  limbs  of 
the  sun  and  comparing  the  mean  of  these  two  watch  readings 
with  the  instant  of  local  mean  time  corresponding  to  0  hours, 
local  apparent  time — the  time  of  transit  of  the  sun's  center. 
The  difference  is  the  watch  correction  to  local  mean  time.  As 
in  the  work  of  the  last  article,  the  correction  to  standard  time 
instead  of  to  local  mean  time  may  be  determined  if  the  longitude 
of  the  place  is  accurately  known. 

Under  ordinary  conditions,  this  observation  is  not  likely  to 
give  quite  as  accurate  results  as  that  of  the  preceding  article. 

Outline  of  Observation: 
Computations  Preceding  Field  Work: 

Compute  local  mean  time  of  transit  of  sun's  center. 

Change  0  hours,  LAT,  for  the  given  date,  to  LMT 
by  method  of  Art.  25,  page  30. 

Field  Work: 

Set  transit  over  one  of  two  stakes  which  mark  a  meridian, 
sight  on  the  other,  and  turn  telescope  to  approxi- 
mately the  meridian  altitude  of  the  sun. 

Note   the  watch  reading  at  the    instant  that  each  limb 

crosses  the  vertical  cross-hair. 
Computations  Following  Field  Work: 

Compute  watch  correction  to  local  mean  time  by  comparing 
mean  of  watch  readings  (taken  at  instants  of  transit 
of  west  and  east  limbs  of  sun)  with  computed  time 
of  transit  of  sun's  center. 


TIME   BY   TRANSIT   OF   THE   SUN  71 

An  example  of  the  computations  and  field  notes  of  this  obser- 
vation is  given  on  pages  124  and  125. 

The  watch  correction  might  be  obtained,  though  probably  less 
accurately,  by  solving  the  astronomical  triangle  for  the  hour 
angle  of  the  sun's  center  from  the  data  obtained  as  described 
in  Art.  46,  page  62,  and  comparing  this  (after  having  changed  it 
from  local  apparent  to  local  mean  time)  with  the  mean  of  the 
watch  readings. 


CHAPTER  X 
OBSERVATIONS  FOR  LONGITUDE 

50.  Longitude  by  Transportation  of  Timepiece.  Since  the 
difference  in  longitude  between  two  places  is  simply  the  differ- 
ence in  local  mean  time,  the  most  practicable  means  for  deter- 
mining longitude  with  field  instruments  is  to  compare  the  local 
mean  time  at  one  of  the  places  (determined  by  one  of  the  obser- 
vations described  in  the  last  chapter)  with  the  reading  of  a 
reliable  watch  or  chronometer  which  is  set  to  the  local  mean 
time  of  the  other  place. 

If  the  watch  is  keeping  standard  time — the  local  mean  time 
at  a  "standard  "  meridian — the  difference  between  the  watch 
reading  and  local  mean  time  as  determined  by  observation  is 
the  difference  in  longitude  of  the  standard  and  local  meridians. 
The  longitude  of  the  standard  meridian  with  respect  to  the 
meridian  of  Greenwich  being  known,  that  of  the  local  meridian 
may  be  obtained. 

This  method  involves  an  approximation  of  the  longitude  for 
use  in  changing  the  sidereal  time  of  transit  of  the  stars  (assuming 
that  local  mean  time  is  to  be  determined  by  star  transits,  as 
outlined  in  Art.  48,  page  68)  to  local  mean  time;  but  the  error 
produced  in  this  computation  by  a  relatively  large  error  in 
assumed  longitude  is  so  small  that  it  is  not  likely  to  be  of  ap- 
preciable amount  in  work  of  this  class.  If  it  is  in  any  case 
sufficiently  large  to  be  considered,  a  second  determination, 
using  for  the  longitude  in  the  second  computation  the  value 
obtained  from  the  first  observation,  will  eliminate  this  difficulty. 

In  precise  geodetic  work,  the  local  mean  time  at  two  stations 
whose  difference  in  longitude  is  being  determined  is  compared 
by  the  electric  telegraph,  elaborate  apparatus  being  used. 

It  is  possible  to  determine  the  longitude — usually  with  rather 
indifferent  precision — by  an  observation  of  the  time  of  transit 
of  the  moon;  but  this  method  has  little  apparent  advantage  over 
that  described  above  of  transportation  of  timepiece,  and  its 
discussion  is  left  to  more  complete  works  on  field  astronomy. 

The  necessity  for  the  determination  of  longitude  arises  very 
seldom  in  general  practice  of  engineering,  for  a  locality  must 

72 


TRANSPORTATION   OF   TIMEPIECE  73 

ordinarily  be  very  remote  whose  longitude  can  not  now  be 
obtained  from  some  map  as  accurately  as  it  can  be  determined 
with  ordinary  field  instruments.  From  a  map  on  which  dis- 
tances can  be  scaled  to  the  nearest  mile,  longitude  can  be  ob- 
tained to  the  nearest  minute  of  arc,  and  a  determination  with 
field  instruments  which  is  accurate  to  the  nearest  half-minute  of 
time  (seven  and  one  half  minutes  of  arc)  is  exceptionally  good 
work. 


CHAPTER  XI 
SUMMARY   OF    OBSERVATIONS 

The  observations  which  have  been  described  in  the  foregoing 
chapters  are  restated  here  in  order  to  summarize,  for  convenient 
reference  when  selecting  a  method  of  observation,  the  data 
required  for  each,  and  to  state  briefly  some  of  their  relative 


When  stating  the  books,  etc.,  required  for  use,  it  is  assumed 
that  the  observer  has  at  hand  a  set  of  logarithmic  and  trigono- 
metric tables. 

51.  Observations  for  Latitude. 

Latitude  by  a  Circumpolar  Star  at  Culmination.  This  is  one 
of  the  most  precise  and  convenient  methods  of  determining 
latitude.  If  Polaris  is  used,  this  book  furnishes  all  necessary 
data.  If  any  other  circumpolar  star  is  used,  either  the  " American 
Ephemeris  and  Nautical  Almanac"  or  the  "American  Nautical 
Almanac"  will  be  needed  from  which  to  obtain  its  right  ascension 
and  declination,  and  in  this  case  an  approximate  value  of  the 
longitude  will  be  needed. 

Latitude  by  Meridian  Altitude  of  a  Southern  Star.  The  pre- 
cision attainable  by  this  method  is  equivalent  to  that  of  the 
preceding  observation,  but  it  is  usually  more  difficult  to  identify 
a  southern  than  a  circumpolar  star.  (See  Fig.  12,  page  37.) 
Either  the  "American  Ephemeris  and  Nautical  Almanac"  or  the 
"American  Nautical  Almanac"  will  be  needed  for  the  right 
ascension  and  declination  of  the  star.  The  longitude  should  be 
known  approximately. 

Latitude  by  Meridian  Altitude  of  the  Sun.  The  precision  of 
this  observation  is  probably  rather  inferior  to  that  of  the  two 
preceding.  It  is  more  convenient  in  that  it  may  be  done  in 
the  daytime.  Either  the  "American  Ephemeris  and  Nautical 
Almanac,"  or  the  "American  Nautical  Almanac,"  or  one  of  the 
pocket  ephemerides  published  by  instrument-makers,  will  furnish 
all  the  required  data  not  given  in  this  book.  The  longitude 
should  be  known  within  a  few  minutes  of  arc. 

52.  Observations  for  Azimuth. 

Azimuth  by  a  Circumpolar  Star  at  Elongation.    This  is  one 
74 


OBSERVATIONS    FOR   AZIMUTH  75 

of  the  most  satisfactory  methods  of  determining  true  azimuth 
under  field  conditions.  Polaris  is  the  best  star  to  use  unless  (for 
a  short  time  in  the  Spring  and  again  in  the  Fall)  its  elongations 
occur  during  daylight.  If  Polaris  is  used,  all  necessary  data, 
with  the  exception  of  the  latitude,  may  be  obtained  from  this 
book.  For  any  other  circumpolar  star  the  "American  Ephemeris 
and  Nautical  Almanac"  or  the  "American  Nautical  Almanac" 
will  be  needed.  The  latitude  of  the  station  should  be  obtained 
to  the  nearest  minute  either  from  a  reliable  map  or  from  a 
previous  observation,  and  in  case  another  star  than  Polaris  is 
to  be  used,  the  longitude  should  be  known  approximately. 

Azimuth  by  Polaris  Near  Elongation.  This  is  a  good  observa- 
tion to  use  when  Polaris  can  not  be  seen  at  elongation,  but  can 
be  seen  within  a  half -hour  of  elongation.  All  necessary  data 
may  be  obtained  from  this  book,  with  the  exception  of  the 
latitude,  which  should  be  obtained  to  the  nearest  minute  from 
a  map  or  from  an  observation,  as  above. 

Azimuth  by  a  Circumpolar  Star  at  Any  Hour  Angle.  This  is  a 
very  precise  method  if  standard  time,  correct  within  a  few 
seconds,  can  be  obtained.  It  permits  a  greater  number  of 
measurements  of  the  direction  of  the  star  than  either  of  the  two 
preceding,  and  may  therefore  be  made  more  precise.  Either 
the  "American  Ephemeris  and  Nautical  Almanac"  or  the  "Ameri- 
can Nautical  Almanac"  will  be  needed  to  obtain  the  right  as- 
cension and  declination  of  the  star,  and  the  latitude  must  be 
obtained  from  a  map  or  a  previous  observation.  An  approxi- 
mate value  of  the  longitude  will  be  needed  in  computing  the 
sidereal  time. 

Azimuth  by  an  Altitude  of  the  Sun.  This  method  has  the 
advantage  that  it  may  be  made  in  the  daytime  and  during  the 
progress  of  a  survey.  Its  precision  is  probably  inferior  to  that 
of  the  observations  on  the  stars.  Either  the  "American 
Ephemeris  and  Nautical  Almanac,"  or  the  "American  Nautical 
Almanac,"  or  one  of  the  pocket  solar  ephemerides  (preferably 
for  Greenwich  mean  noon)  published  by  the  instrument  makers, 
will  furnish  the  required  declination  of  the  sun.  The  latitude 
and  longitude  should  be  obtained,  the  former  to  the  nearest 
minute  from  a  map  or  a  previous  observation,  the  latter  less 
accurately  from  a  map.  Any  other  required  data  will  be  found 
in  this  book. 

Azimuth  by  Equal  Altitudes  of  a  Star.  This  observation  may 
be  made  to  give  a  very  high  degree  of  precision,  and  no  tables 


76  SUMMARY   OF   OBSERVATIONS 

or  data  whatever  are  required.  It  is  inconvenient  in  that  it 
requires  two  series  of  observations  to  be  made  at  an  interval 
of  from  four  to  six  hours. 

63.  Observations  for  Time. 

Time  by  Transit  of  Star.  This  is  one  of  the  simplest  and  best 
methods  of  determining  local  time.  Data  from  the  " American 
Nautical  Almanac,"  or  preferably  from  the  " American  Ephemeris 
and  Nautical  Almanac,"  are  required.  The  latitude  and  longi- 
tude should  be  known  approximately.  It  is  necessary  that  the 
direction  of  the  meridian  should  be  marked  on  the  ground. 

Time  by  Transit  of  Sun.  This  method  is  probably  inferior  in 
accuracy  to  the  preceding,  but  it  is  more  convenient  in  that  it 
may  be  made  in  the  daytime.  Either  the  " American  Ephemeris 
and  Nautical  Almanac,"  or  the  "American  Nautical  Almanac," 
or  one  of  the  pocket  solar  ephemerides  (preferably  for  Greenwich 
mean  noon)  published  by  instrument-makers,  will  furnish  re- 
quired data.  The  longitude  must  be  known  approximately. 

64.  Observations  for  Longitude. 

Longitude  by  Transportation  of  Timepiece.  Since  the  deter- 
mination of  longitude  by  this  method  consists  essentially  in 
the  determination  of  local  mean  time,  the  remarks  above  in 
regard  to  observations  for  time  apply  here  as  well. 

It  should  be  apparent  that  if  a  determination  of  latitude, 
longitude,  azimuth,  and  time  were  to  be  made  at  a  station  where 
all  four  were  entirely  unknown,  values  of  some  of  the  quantities 
would  have  to  be  approximated  in  making  observations  for  the 
others  and  the  values  obtained  from  the  observations  used  for 
a  closer  approximation,  until  by  a  series  of  observations  the 
values  of  all  were  obtained  to  the  required  degree  of  refinement. 


APPENDIX  A 
SPHERICAL  TRIGONOMETRY 

Derivation  of  Formulas  Required  in  Field  Astronomy 

A  portion  of  the  surface  of  a  sphere  bounded  by  arcs  of  three 
great  circles  is  called  a  spherical  triangle. 

If  the  vertices  of  the  spherical  triangle  be  joined  by  straight 
lines  to  the  center  of  the  sphere,  a  triedral  angle  is  formed.  The 
face  angles  of  the  triedral  angle  measure  the  sides  of  the  spherical 


FIG.  21. 

triangle,  and  the  diedral  angles  of  the  triedral  angle  are  measures 
of  the  angles  of  the  spherical  triangle.  Both  sides  and  angles 
of  a  spherical  triangle  are  usually  measured  in  degrees. 

The  fundamental  formulas  required  for  the  solution  of  spherical 
triangles  may  be  derived  by  the  methods  of  analytic  geometry, 
as  follows: 

In  Fig.  21: 

Let:  ABC  be  a  spherical  triangle  on  the  surface  of  a 

sphere  whose  center  is  at  0. 

Then:  OA  =  OB  =  OC  =  r. 

77 


78 


Assume: 


Draw: 


APPENDIX   A 

0  as  origin  of  rectangular  coordinates. 
OX  through  vertex  A. 
OF  in  plane  of  AOB. 
OZ  perpendicular  to  plane  of  AOB. 

CP  perpendicular  to  plane  of  OX  and  OF. 
PS  perpendicular  to  OX. 

CS. 


Then :     Coordinates  of  C  are : 

x  =  OS,  y  =  PS,  z  =  PC. 

By  construction: 

Angle  CSP  =  spherical  angle  A, 
Angle  COS  =  side  b. 

Then:  x  =  r  •  cos  &,  y  =  r  •  sin  b  •  cos  A, 

z  =  r  •  sin  b  •  sin  A. 


V 

Y 

V*: 

N 

a; 

u\"" 

\ 

""";:r"^-ixp 

„-"/        v         '» 

\~  -" 

«'        \        '\ 

M^*"""" 

Y 

»       '  \ 

\ 

\       '    \ 

\      '    \ 

y' 

-£•' 

\ 
\ 

\    j" 

tl 

X 

-  " 

lO 

S 

\ 

FIG.  21a. 


Leaving  OZ  unchanged,  revolve  OX  and  OF  to  positions 
OX'  and  OF'  (Fig.  21a)  i.e.,  until  OX' 
passes  through  vertex  B. 

Now,  if  from  P  a  line  were  drawn  perpendicular  to  OX',  and 
its  point  of  intersection  with  OX'  were 
joined  to  C,  we  should  have: 
x'  =  r  •  cos  a,  2/'  =  —  r  •  sin  a  •  cos  B, 
2'  =  r  •  sin  a  •  sin  B. 

The  equations  of  transformation  of  coordinates  in  this  case  are : 
x'  =  y  •  sin  c  •  +  z  •  cos  c,    y'  =  y  •  cos  c  —  #  •  sin  c,    2'  =  2. 


SPHERICAL   TRIGONOMETRY 


79 


Substituting  in  these  equations  the  values  of  x,  y,  z,  2',  y',  z', 
and  dividing  by  r,  we  have: 

cos  a  =  cos  b  •  cos  c  +  sin  b  •  sin  c  •  cos  A 

sin  a  •  cos  B  =  cos  b  •  sin  c  —  sin  b  •  cos  c  •  cos  A 

sin  a  •  sin  B  =  sin  b  •  sin  A 

which  are  the  three  Fundamental  Laws  of  Spherical  Trigonometry. 

Since  we  are  at  present  con- 
cerned with  solutions  of  the  astro- 
nomical triangle,  lettered  as  shown 
in  Fig.  22  and  in  other  figures 
which  show  the  celestial  sphere, 
it  will  be  convenient  to  have 
the  notation  of  the  formulas 
changed  to  correspond,  as 
follows:  ,-, 


FIG.  22. 


cos  s  =  cos  p  •  cos  z  +  sin  p  •  sin  z  •  cos  S  .  .  .  (1) 
sin  s  •  cos  P  =  xx>s  p  •  sin  z  —  sin  p  •  cos  z  •  cos  S.  (2) 
sin  s  •  sin  P  =  sin  S  •  sin  p (3) 

From  these  fundamental  formulas  others  may  be  derived 
which  are  more  convenient  for  special  solutions  of  the  astronomi- 
cal triangle. 

To  express  the  sines,  cosines,  and  tangents  of  the  half-angles 
of  the  astronomical  triangle  in  terms  of  functions  of  the  sides: 
From  (1) : 

cos  z  =  cos  s  •  cos  p  +  sin  s  •  sin  p  •  cos  Z 
Whence: 

cos  z  —  cos  s  •  cos  p 


cosZ  = 


sin  s  •  sin  p 


(a) 


Subtracting  both  sides  from  1 : 


1  -  cos  Z  =  I  - 


cos  z  —  cos  s  •  cos  p 


sm  s  -  sin  p 

sin  s  -  sin  p  +  cos  s  -  cos  p  —  cos  z 
sin  s  •  sin  p 

From  plane  trigonometry: 

cos  (  A  —  B  )  =  cos  A  -  cos  B  +  sin  A  •  sin  5 
and:  2  sin2  A/2  =  1  -  cos  A 


80  APPENDIX   A 

Substituting: 

Z        cos  (s  —  p)  —  cos  z 

2  sm2  —  = ; ; 

2  sm  s  •  sm  p 

From  plane  trigonometry : 

A  +  B      .    A  -  B 

cos  B  —  cos  A  =  2  sm •  sm .      .      (b) 

—i  2i 

Whence: 

Z        2  sin  3/£  [z  +  (s  —  p)]  •  sin  %[z  —  (s  —  p)] 

2  sin2  —  = : — 

2  sm  s  •  sm  p 

Z        sin  y^  (z  +  s  —  p)  •  sin  K  (z  —  «  +  P) 

sm2  —  =  : : 

2  sm  s  •  sm  p 

Let:  2k  =  s  +  p+z  =  270°  -  (0  +  h  +  5) 

Then:        z  +  s  -  p  =  (s  +  p  +  z)  -  2p  =  2k  -  2p 
z  -  s  +  p  =  (s  +  p  +  z)  -  2s    =  2k  -  2s 

Substituting: 

Z        sin  (k  —  s)  •  sin  (k  —  p) 

sm2  —  =  - 


sin  s  •  sin  p 


Z  I  sir 

'T  =  \- 


~  sin  (k  —  s)  •  sin  (k  —  p)  ,  , 

sm  —  =  \  r-    — .      •       (c) 

sm  s  •  sm  p 

Substituting  <j>  =  90°  -  s,  h  =  90°  -  p,  d  =  90°  -  z,  in  Equa- 
tion (c): 


Z  cos  (k  +  0)  •  cos  (k -j-h) 

sin  —  =  A .     .      (4) 

cos  0  •  cos  h 


In  like  manner: 


P  /sir 

•T  =  \- 


Jsin  (/c  —  s)  •  sin  (k  —  z) 
: : 

sm  s  •  sm  z 


P  I  cos  (k 

LT  =  \ — ; 


</>)  "  cos  (k  +  3) 


cos  <f>  •  cos  8 
Adding  both  members  of  Equation  (a)  to  1 : 
cos  z  —  cos  s  •  cos  p 


cos  Z  =  1 


sm  s  •  sm  p 
cos  z  —  (cos  s  •  cos  p  —  sin  s  •  sin  p) 
sin  s  •  sin  p 


SPHERICAL   TRIGONOMETRY  81 

From  plane  trigonometry: 

cos  (A  +  B)  —  cos  A  •  cos  B  —  sin  A  •  sin  B 

A 
and  2  cos2  —  =  1  +  cos  A 

4B 

Substituting: 

Z       cos  z  —  cos  (s  +  p) 

2  cos2  —  =  : : 

2  sin  s  •  sin  p 

Whence,  by  equation  (b) : 

Z        2  sin  K  (s  +  P  +  z)  -  sin  Y2  (s  +  p  -  g) 

2  cos2  —  = ; 

2  sin  s  •  sin  p 

Again  putting  2k  =  s  +  P  +  2,  whence:    s  +  P  —  2  =  2k  —  2z: 

Z        sin  k  •  sin  (fc  —  z) 

cos2  —  =  : ; 

2  sin  s  •  sin  p 


cos 


Z  /  sin  k  •  sin  (k  —  z) 

2          if       sin  s  •  sin  p 


Z  I  sin  k  •  cos  (k  +  a)                              ,flN 

or                cos  —  =  \  -  "  —       ....      (6) 

2  \       cos  0  •  cos  h 

In  like  manner: 

P  /  sin  k  •  sin  (k  —  p) 

cos  —  =  \l  -  ;  -  :  - 

2  \         sin  s  •  sin  z 

P  /  sin  k  •  cos  (k  +  h) 

or                cos  —  =  -\l  ""1  -  "      ....      (7) 

2  \        cos  0  •  cos  8 


Dividing  (4)  by  (6)  : 


Z          I  cos  (k  +  </>)  •  cos  (k  +  h)        I       cos  <f>  -  cos  h 
2         \  cos  </>  •  cos  /i  \  sin  fc  •  cos  (&  +  6) 

Z  |  cos  (k  +  0)  •  cos  (k  +  h) 

^"~  —  A  I  ^  ^  ~  ^~"  ~~  ""  ~~  '^^^™™  .      (o 

2        Af        sin  k  •  cos  (k  +  5) 


tan 
In  like  manner: 


tan        . 


2        \       sin  k  •  cos  (k  +  h) 


82  APPENDIX   A 

A  convenient  formula  for  computing  the  angle  Z  when  the 
known  data  are  t,  <£,  and  6,  may  be  derived  as  follows: 
From  (3): 

sin  p  •  sin  Z  =  sin  P  •  sin  z  .      .      .      .      (d) 
From  (2): 

sin  p  •  cos  Z  =  cos  z  •  sin  s  —  sin  z  •  cos  s  •  cos  P  .    (e) 
Dividing  (d)  by  (e): 

sin  P  •  sin  z 

tan  Z  = — 

cos  z  •  sin  s  —  sin  z  •  cos  s  •  cos  P 

sin  P 

cos  z  •  sin  s  —  cos  s  •  cos  P 

sin  t 

or  tan  Z  = .      .    (10) 

tan  d  -  cos  $  —  sin  0  •  cos  t 

A  formula  for  computing  Z  when  5,  t,  and  h  are  known,  may 
be  derived  directly  from  (3) : 

sin  p  '  sin  Z  =  sin  P  .  sin  z 
Whence: 

sin  P  •  sin  z 


sinZ 


sin  p 


sin  t  •  cos  5 

or  sin  Z  = (11) 

cos  h 

The  formula  for  Z  when  the  astronomical  triangle  is  right- 
angled  at  S,  comes  directly  from  (3) : 

sin  s  •  sin  Z  =  sin  S  •  sin  z 

Since  S  =  90°,  sin  S  =  1,  and: 

.  sin  z 

sin  Z  =  — — 

sin  s 

cos  8 

or  sin  Z  = (12) 

cos  0 

The  formula  for  P  under  similar  conditions  (S  =  90°)  may 
be  derived  as  follows: 
From  (2) : 

sin  s  •  cos  P  =  cos  p  •  sin  z  —  sin  p  •  cos  z  •  cos  S 
Since  S  =  90°,  cos  S  =  0,  and: 

sin  s  •  cos  P  =  cos  p  •  sin  z 


SPHERICAL   TRIGONOMETRY  83 

Whence: 

cos  p  •  sin  z 
cos  P  = -. • (f) 

sin  s 

From  (1): 

cos  s  =  cos  p  •  cos  z  +  sin  p  •  sin  z  •  cos  S 
Since  S  =  90°,  cos  S  =  0,  and: 

cos  s  =  cos  p  •  cos  z 

Multiplying  the  numerator  of  (f)  by  (cos  s),  and  the  denominator 
by  its  equal  (cos  p  •  cos  z) : 

cos  p  •  sin  z  cos  s 

cos  P  = 


sin  s  cos  p  •  cos  z 

tan  z 
tan  s 


or  cosP  =  ......    (13) 

tan  5 


APPENDIX  B 
SOLAR  ATTACHMENTS  FOR  TRANSITS 

The  Solar  Attachment  is  a  device  which  is  mounted  upon  or 
beside  the  telescope  of  an  engineer's  transit,  and  is  used  chiefly 
for  direct  determination  of  the  true  meridian  by  an  observation 
on  the  sun.  By  its  use  the  astronomical  triangle  is  solved  me- 
chanically, and  at  the  end  of  an  observation  the  line  of  sight 
through  the  transit  telescope  should  lie  in  the  plane  of  the 
meridian. 

There  are  several  different  forms  of  the  solar  attachment 
made,  but  they  are  all  alike  in  principle  and  differ  only  slightly 
in  method  of  use.  The  essential  features  of  all  are:  a  polar 
axis  which  is  perpendicular  to  the  plane  defined  by  the  horizontal 
axis  of  the  transit  and  the  line  of  sight  of  the  transit  telescope; 
a  small  telescope,  called  the  solar  telescope,  which  is  so  mounted 
as  to  revolve  about  the  polar  axis,  and  which  may  also  be  set 
at  any  desired  inclination  to  the  plane  of  the  horizontal  axis 
and  line  of  sight  of  the  main  telescope;  and  a  declination  arc 
or  other  means  of  measuring  this  inclination. 

In  one  well-known  form — the  Burt  Solar  Attachment,  shown 
in  Fig.  23,  page  85 — the  solar  telescope  is  replaced  by  a  small 
lens  and  a  silver  screen  on  which  the  sun's  image  may  be  thrown, 
thus  defining  a  line  of  sight  to  the  sun.  On  this  form  of  the 
instrument  there  is  a  declination  arc  on  which  may  be  set  off 
any  desired  inclination  of  the  line  of  sight  to  the  sun  with  respect 
to  the  plane  defined  by  the  horizontal  axis  and  telescope  of 
the  transit. 

Another  common  form  of  the  attachment  is  that  shown  in 
Fig.  24,  page  86.  It  has  a  telescope  for  determining  the  line 
of  sight  to  the  sun;  but  the  inclination  of  this  line  of  sight  to 
the  plane  of  the  transit  telescope  and  horizontal  axis  is  deter- 
mined by  means  of  the  small  level  tube  which  is  mounted  on 
the  solar  telescope,  used  in  connection  with  the  vertical  arc  or 
circle  of  the  transit.  This  inclination  may  be  set  off  in  the 
following  manner: 

84 


SOLAR   ATTACHMENTS    FOR   TRANSITS  85 


FIG.  23.    THE  BURT  SOLAR  ATTACHMENT. 


86 


APPENDIX   B 


First  bring  the  solar  and  transit  telescopes  into  the  same 
vertical  plane  by  sighting  both  at  a  distant  point. 

Set  off  the  desired  inclination  on  the  vertical  circle,  applying 
index  correction  if  necessary,  and  make  the  solar  telescope 
horizontal  by  means  of  its  own  level. 

The  angle  between  the  two  telescopes  is  now  equal  to  that 
set  off  on  the  vertical  circle. 

The  objective  end  of  the  telescope  should  be  depressed  if  an 
inclination  of  the  line  of  sight  of  the  solar  telescope  above  the 


FIG.  24.    THE  SAEGMULLEB  SOLAR  ATTACHMENT. 

plane  defined  by  the  transit  telescope  and  horizontal  axis  is 
desired  (as  for  north  declinations),  and  should  be  elevated  for 
inclinations  below  that  plane  (as  for  south  declinations). 

To  understand  the  principle  on  which  the  use  of  the  solar 
attachment  is  based,  refer  to  Fig.  25,  page  87.  Suppose  the 
transit  telescope  to  be  so  inclined  that  the  plane  denned  by  the 
line  of  sight  and  the  horizontal  axis  of  the  transit  coincides  with 
the  plane  of  the  equatoi,  and  the  transit  to  be  turned  so  that 
the  line  of  sight  lies  in  the  plane  of  the  meridian.  The  polar  axis 
will  now  coincide  with  the  axis  of  rotation  of  the  earth,  produced; 
i.e.,  it  will  point  toward  the  north  celestial  pole.  Now,  assuming 
for  the  moment  that  the  sun's  declination  remains  constant  for 


SOLAR  ATTACHMENTS   FOR  TRANSITS 


87 


a  day,  and  that  the  solar  telescope  is  inclined  to  the  plane  of  the 
equator  by  an  amount  equal  to  this  declination,  the  sun's  motion 
may  be  followed  by  the  solar  telescope  by  simply  revolving  the 
latter  about  the  polar  axis,  without  disturbing  the  rest  of  the 
instrument.  Since  the  declination  of  the  sun  does  change,  its 
motion  cannot  be  followed  throughout  the  day  without  changing 


North  I 


*South 


the  inclination  of  the  solar  telescope;  but  it  can  be  practically 
followed  for  twenty  or  thirty  minutes,  even  at  the  seasons  of 
most  rapid  change  in  declination— in  June  and  December. 

This  cannot  be  done  unless  the  polar  axis  coincides  with  the 
axis  of  the  celestial  sphere  (points,  in  the  northern  hemisphere, 
to  the  north  celestial  pole)  and  the  telescope  lies  in  the  plane 
of  the  meridian.  This  condition  furnishes  the  principle  on 
which  the  work  of  an  observation  is  based. 

The  declination  of  the  sun,  computed  for  the  time  at  which 
the  observation  is  to  be  made,  must  be  corrected  for  refraction. 
The  accuracy  with  which  the  declination  can  be  set  off  does  not 


88  APPENDIX   B 

warrant  the  use  of  a  parallax  correction.  To  obtain  the  "ap- 
parent" from  the  computed  or  true  declination,  the  refraction 
correction  is  always  added  algebraically,  regarding  north  declina- 
tions as  plus  and  south  declinations  as  minus.  The  amount  of 
this  correction  may  be  obtained  from  Table  III,  page  91, 
using  as  the  altitude  the  approximate  measured  altitude  of  the 
sun;  or  it  may  be  obtained  more  easily  from  tables  such  as  are 
published  by  the  different  instrument-makers  for  use  with  solar 
attachments;  in  which  the  .refraction  corrections  are  given  for 
various  latitudes,  declinations,  and  hour  angles  of  the  sun  (before 
or  after  transit). 

The  observation  for  azimuth  with  the  solar  attachment  may 
now  be  outlined  as  follows: 

Set  up  the  transit  over  a  centered  stake. 

Incline  the  solar  telescope  to  the  plane  of  the  transit  telescope 
and  horizontal  axis  by  an  amount  equal  to  the  declination  of 
the  sun — "  corrected  "  for  refraction — computed  for  the  day 
and  hour  of  the  observation. 

Make  sure  that  the  transit  is  accurately  leveled. 

Without  disturbing  the  relative  inclination  of  the  two  tele- 
scopes incline  the  transit  telescope  by  an  amount  equal  to  the 
co-latitude  of  the  place  (making  the  polar  axis  point  toward  the 
north  celestial  pole);  applying  index  correction,  if  necessary,  to 
obtain  the  proper  setting  for  the  vertical  arc  or  circle. 

By  revolving  the  transit  about  its  vertical  axis  and  the  solar 
telescope  about  the  polar  axis,  bring  the  sun's  image  into  the 
center  of  the  square  formed  by  the  four  cross-hairs  of  the  solar 
telescope  or  by  the  four  lines  on  the  silver  plate  of  the  attach- 
ment which  has  no  telescope.  Finish  the  setting  with  the 
tangent  screws.  A  magnifying  glass  should  be  used  in  check- 
ing the  position  of  the  image  on  the  silver  plate. 

Only  one  position  can  be  found  in  which  this  setting  can  be 
properly  made.  When  this  position  is  found  it  should  be  pos- 
sible to  follow  the  motion  of  the  sun  for  several  minutes,  keeping 
the  image  in  the  little  square,  by  simply  turning  the  solar  tele- 
scope about  the  polar  axis,  and  without  otherwise  disturbing 
the  instrument. 

The  line  of  sight  of  the  transit  telescope  should  now  be  in  the 
plane  of  the  meridian;  and  a  stake  may  be  lined  in  and  centered, 
which,  with  the  stake  under  the  instrument,  will  define  the  direc- 
tion of  the  meridian. 

Also,  the  approximate  local  apparent  time  should  now  be 


SOLAR  ATTACHMENTS  FOR  TRANSITS        89 

indicated  on  a  graduated  circle  which  is  near  the  base  of  the 
polar  axis  on  some  types  of  the  instrument,  and  is  perpendicular 
to  the  polar  axis. 

Most  of  the  companies  making  solar  attachments  publish 
yearly  a  little  pamphlet  containing,  in  addition  to  a  solar 
ephemeris  and  other  useful  tables,  detailed  directions  for  the 
adjustment  and  use  of  their  own  instruments.  Some  com- 
panies publish  these  directions  in  a  pamphlet  or  book  separate 
from  their  solar  ephemeris.  The  reader  is  referred  to  these 
publications  for  more  particular  information  in  regard  to  the 
several  solar  attachments  now  on  the  market. 

Observations  for  latitude  and  for  time  may  be  made  with  the 
solar  attachment,  but  it  is  believed  that  its  use  in  these  observa- 
tions presents  but  little  advantage  over  the  methods  given  in 
Chapters  VII  and  IX. 

The  chief  advantage  of  the  solar  attachment  in  observations 
for  azimuth  is  that  the  direction  of  the  meridian  may  be  obtained 
directly,  at  any  time  of  day  when  the  sun  is  visible  and  with  a 
minimum  of  computation.  For  good  results,  however,  observa- 
tions should  not  be  made  when  the  altitude  of  the  sun  is  less 
than  10°  or  15°  or  within  an  hour  of  noon. 

The  solar  attachment  may  also  be  used  in  a  series  of  observa- 
tions for  running  lines  by  true  bearings  much  as  a  compass  is 
used  for  running  lines  by  magnetic  bearings.  This  work  is 
explained  in  the  publications  of  the  instrument-makers  and  in 
most  texts  on  plane  surveying. 

In  regard  to  the  accuracy  attainable,  it  is  claimed  by  some 
that  results  accurate  within  a  quarter  of  a  minute  may  be 
obtained  in  observations  for  azimuth  with  the  solar  attachment. 
It  is  believed,  however,  that  the  nearest  minute  of  arc  represents 
about  as  high  a  degree  of  accuracy  as  is  likely  to  be  consistently 
realized. 


TABLES 


SIDEREAL   INTO   MEAN   SOLAR   TIME 


93 


TABLE  I 

CONVERSION  OP  SIDEREAL  INTO  MEAN  SOLAR  TIME 
Corrections  to  be  Subtracted  from  a  Sidereal  Time  Interval 


Sid. 
Hrs. 

Corr. 
m        s 

Sid. 
Min. 

Corr. 

s 

Sid. 
Min. 

Corr. 

8 

Sid. 
Sec. 

Corr. 
s 

Sid. 
Sec. 

Corr. 

8 

1  

0     09.83 

1 

0.16 

31 

5.08 

1 

0.003 

31 

0.08 

2  ..    . 

0     19.66 

2 

0.33 

32 

5.24 

2 

0.005 

32 

0.09 

3  
4  
5  

0     29.49 
0     39.32 
0     49.15 

3 

4 
5 

0.49 
0.66 
0.82 

33 
34 
35 

5.41 
5.57 
5.73 

3 

4 
5 

0.008 
0.011 
0.014 

33 
34 
35 

0.09 
0.09 
0.10 

6  
7  

0     58.98 
1     08.81 

6 

7 

0.98 
1.15 

36 
37 

5.90 
6.06 

6 

7 

0.02 
0.02 

36 
37 

0.10 
0.10 

8     . 

1     18.64 

8 

1.31 

38 

6.23 

8 

0.02 

38 

0.10 

9... 

1     28.47 

9 

1.47 

39 

6.39 

9 

0.03 

39 

0.11 

10     . 

1     38  30 

10 

1.64 

40 

6.55 

10 

0.03 

40 

0.11 

11  
12  
13  

1     48.13 
1     57.96 
2     07.78 

11 
12 
13 

1.80 
1.97 
2.13 

41 
42 
43 

6.72 
6.88 
7.05 

11 

12 
13 

0.03 
0.03 
0.04 

41 
42 
43 

0.11 
0.12 
0.12 

14     . 

2     17  61 

14 

2  29 

44 

7  21 

14 

0.04 

44 

0.12 

15  

2     27  44 

15 

2.46 

45 

7.37 

15 

0.04 

45 

0.12 

16 

2     37  27 

16 

2  62 

46 

7  54 

16 

0  04 

46 

0  13 

17... 

2     47  10 

17 

2  79 

47 

7  70 

17 

0  05 

47 

0.13 

18  
19... 

2     56.93 
3     06  76 

18 
19 

2.95 
3  11 

48 
49 

7.86 
8  03 

18 
19 

0.05 
0  05 

48 
49 

0.13 
0.13 

20  
21... 

3     16.59 
3     26  42 

20 
21 

3.28 
3  44 

50 
51 

8.19 
8  36 

20 
21 

0.06 
0  06 

50 
51 

0.14 
0  14 

22     . 

3     36  25 

22 

3  60 

52 

8  52 

22 

0  06 

52 

0  14 

23... 

3     46  08 

23 

3  77 

53 

8  68 

23 

0  06 

53 

0  15 

24 

3     55  91 

24 

3  93 

54 

8  85 

24 

0  07 

54 

0  15 

25 
26 
27 
28 
29 
30 

4.10 
4.26 
4.42 
4.59 
4.75 
4.92 

55 
56 
57 
58 
59 
60 

9.01 
9.17 
9.34 
9.50 
9.67 
9.83 

25 
26 
27 
28 
29 
30 

0.07 
0.07 
0.07 
0.08 
0.08 
0.08 

55 
56 
57 
58 
59 
60 

0.15 
0.15 
0.16 
0.16 
0.16 
0.16 

MEAN    SOLAR   INTO    SIDEREAL   TIME 


TABLE  II 

CONVERSION  OF  MEAN  SOLAR  INTO  SIDEREAL  TIME 
Corrections  to  be  Added  to  a  Mean  Solar  Time  Interval 


Mean 
Solar 
Hrs. 

Corr. 
m         s 

Mean 
Solar 
Min. 

Corr. 

s 

Mean 
Solar 
Min. 

Corr. 

s 

Mean 
Solar 
Sec. 

Corr. 

s 

Mean 
Solar 
Sec. 

Corr. 

s 

1... 

0     09.86 

1 

0.16 

31 

5  09 

1 

0  003 

31 

0  09 

2  

0     19  71 

2 

0  33 

32 

5  26 

2 

0  005 

32 

0  09 

3  

0     29  .  57 

3 

0.49 

33 

5.42 

3 

0.008 

33 

0  09 

4  .. 

0     39  43 

4 

0  66 

34 

5  59 

4 

0  Oil 

34 

0  09 

5... 

0     49.28 

5 

0.82 

35 

5.75 

5 

0.014 

35 

0  10 

6  .. 

0     59  14 

6 

0  99 

36 

5  91 

6 

0  02 

36 

0  10 

7  

1     09.00 

7 

1.15 

37 

6.08 

7 

0.02 

37 

0  10 

8   ..   . 

1     18  85 

8 

1  31 

38 

6  24 

8 

0  02 

38 

0  10 

9  

1     28.71 

9 

1.48 

39 

6.41 

9 

0.03 

39 

0  11 

10  .. 

1     38  57 

10 

1  64 

40 

6  57 

10 

0  03 

40 

0  11 

11... 

1     48.42 

11 

1.81 

41 

6.74 

11 

0.03 

41 

0  11 

12     . 

1     58  28 

12 

1  97 

42 

6  90 

12 

0  03 

42 

0  12 

13  

2     08  .  13 

13 

2.14 

43 

7.06 

13 

0.04 

43 

0  12 

14     . 

2     17  99 

14 

2  30 

44 

7  23 

14 

0  04 

44 

0  12 

15  

2     27.85 

15 

2.46 

45 

7.39 

15 

0.04 

45 

0.12 

1-6  .. 

2     37  70 

16 

2  63 

46 

7  56 

16 

0  04 

46 

0  13 

17  

2     47  .  56 

17 

2.79 

47 

7.72 

17 

0.05 

47 

0  13 

18  
19 

2     57.42 
3     07  27 

18 
19 

2.96 
3  12 

48 
49 

7.89 
8  05 

18 
19 

0.05 
0  05 

48 
49 

0.13 
0  13 

20  

3     17  13 

20 

3  29 

50 

8  21 

20 

0  06 

50 

0  14 

21 

3     26  99 

21 

3  45 

51 

8  38 

21 

0  06 

51 

0  14 

22... 

3     36  84 

22 

3  61 

52 

8  54 

22 

0  06 

52 

0  14 

23 

3     46  70 

23 

3  78 

53 

8  71 

23 

0  06 

53 

0  15 

24  

3     56  .  56 

24 

3  94 

54 

8  87 

24 

0  07 

54 

0  15 

25 
26 
27 
28 
29 
30 

4.11 
4.27 
4.44 
4.60 
4.76 
4.93 

55 
56 
57 
58 
59 
60 

9.04 
9.20 
9.36 
9.53 
9.69 
9.86 

25 
26 
27 
28 
29 
30 

0.07 
0.07 
0.07 
0.08 
0.08 
0.08 

55 
56 
57 
58 
59 
60 

0.15 
0.15 
0.16 
0.16 
0.16 
0.16 

MEAN   REFRACTION 


95 


TABLE  III 

MEAN  REFRACTION 

Corrections  to  be  Subtracted  from  Apparent  (Observed)  Altitudes 
Barometer:  29.6  inches  Temperature:  49°  F. 


Apparent 
Altitude 

Mean 
Refraction 

Apparent 
Altitude 

Mean 
Refraction 

10° 

5'  16" 

38° 

'  14" 

11 

4   49 

39 

11 

12 

4   25 

40 

09 

13 

4   05 

41 

06 

14 

3   47 

42 

04 

15 

3   32 

43 

02 

16 

3   19 

44 

00 

17 

3   07 

45 

0   58 

18 

2   56 

46 

0   56 

19 

2   46 

47 

0   54 

20 

2   37 

48 

0   52 

21 

2   29 

49 

0   50 

22 

2   22 

50 

0   48 

23 

2   15 

51 

0   47 

24 

2   09 

52 

0   45 

25 

2   03 

53 

0  44 

26 

1   58 

54 

0   42 

27 

1   53 

55 

0   40 

28 

1   48 

56 

0   39 

29 

1   44 

57 

0   38 

30 

1   40 

58 

0   36 

31 

1   36 

59 

0   35 

32 

1   32 

60 

0   33 

33 

1   29 

65 

0   27 

34 

1   25 

70 

0   21 

35 

1   22 

75 

0   16 

36 

1   19 

80 

0   10 

37 

1   17 

85 

0   05 

90 

0   00 

For  temperatures  other  than  49°  F.,  the  mean  refraction  may  be  multiplied 
by  the  following  factors: 

For  20°:  1.060,  for  40°:  1.017,  for  60°:  0.978,  for  80°:  0.942,  for  88°:  0.929. 

These  mean  refractions  are  based  on  Bessel's  "  Refraction 
Tables." 


96 


SUN  S   PARALLAX   AND   SEMI-DIAMETER 


TABLE  IV 

SUN'S  PARALLAX  AND  SEMI-DIAMETER 

SUN'S  PARALLAX 


Altitude 

Parallax 

Altitude 

Parallax 

0° 

9" 

50° 

6" 

10 

9 

60 

4 

20 

8 

70 

3 

30 

8 

80 

2 

40 

7 

90 

0 

SUN'S  SEMI-DIAMETER 


Date 

Semi-Diameter 

Date 

Semi-Diameter 

January  1  

16'     18" 

July  1  

15'     46" 

February  1  

16      16 

August  1 

15      47 

March  1    . 

16      10 

September  1 

15      53 

April  1  

16      02 

October  1 

16      01 

May  1  

15      54 

November  1 

16      09 

June  1  

15      48 

December  1  

16      15 

CULMINATIONS   AND   ELONGATIONS   OF   POLARIS      97 


TABLE  V 

LOCAL  MEAN  (ASTRONOMICAL)  TIME  OF  THE  CULMINATIONS 
AND  ELONGATIONS  OF  POLARIS  IN  THE  YEAR  1915 

With  Corrections  for  Referring  the  Tabular  Quantities  to  Other  Years 
(Computed  for  latitude  40°  north  and  longitude  90°  or  6  h  west  of  Greenwich) 


Date 
1915 

East 
Elongation 
h        m 

Upper 
Culmination 
h         m 

West 
Elongation 
h        m 

Lower 
Culmination 
h         m 

January  1  

0     51.7 

6     46.9 

12     42.1 

18     44.9 

January  15 

23     52  5 

5     51  6 

11     46  8 

17     49  6 

February  1  

22     45.3 

4     44.5 

10     39.7 

16     42  5 

February  15 

21     50  1 

3     49  2 

9     44  4 

15     47  2 

March  1     

20     54.8 

2     54.0 

8     49.2 

14     52  0 

March  15  
April  1  

19     59.6 
18     52.7 

1     58.8 
0     51.9 

7     54.0 
6     47.1 

13     56.8 
12     49.9 

April  15  

17     57.7 

23     52.9 

5     52.0 

11     54.8 

May  1      

16     54.8 

22     50.0 

4     49.2 

10     52  0 

May  15 

15     59  9 

21     55  1 

3     54  2 

9     57  0 

June  1      

14     53.3 

20     48.5 

2     47.6 

8     50  4 

June  15 

13     58  5 

19     53  7 

1     52  8 

7     55  6 

July  1  

12     55.9 

18     51.1 

0     50.2 

6     53.0 

July  15 

12     01  1 

17     56  3 

23     51  5 

5     58  2 

August  1    

10     54  5 

16     49.7 

22     44.9 

4     51  7 

August  15  

9     59.8 

15     55.0 

21     50.2 

3     56.9 

September  1    

8     53  2 

14     48.4 

20     43.6 

2     50  3 

September  15  

7     58.3 

13     53.5 

19     48.7 

1     55.4 

October  1  

6     55.5 

12     50.7 

18     45.9 

0     52.7 

October  15  

6     00.6 

11     55.8 

17     51.0 

23     53.8 

November  1  
November  15  
December  1  
December  15 

4     53.7 
3     58.6 
2     55.6 
2     00  4 

10     48.9 
9     53.8 
8     50.8 
7     55  6 

16     44.1 
15     49.0 
14     46.0 
13     50  8 

22     46.9 
21     51.8 
20     48.8 
19     53  6 

Corrections  on  pages  98  and  99 


98      CULMINATIONS    AND    ELONGATIONS    OF    POLARIS 


A.  To  refer  the  above  tabular  quantities  to  years  other  than 
1915: 


For  year  1916, 
1916, 
1917, 
1918, 
1919, 
1920, 
1920, 
1921, 
1922, 
1923, 
1924, 
1924, 
1925, 
1926, 
1927, 
1928, 
1928, 


add 

subtract 

subtract 

add 

add 

add 

add 

add 

add 

add 

add 

add 

add 

add 

add 

add 

add 


m 

1.6  up 
2.3  on 
0.7 
0.9 
2.5 

4.0  up 
0.1  on 
1.6 
3.1 
4.5 

5.9  up 
2.0  on 
3.3 
4.6 
5.9 

7.2  up 

3.3  on 


to  March  1 

and  after  March  1 


to  March  1 

and  after  March  1 


to  March  1 

and  after  March  1 


to  March  1. 

and  after  March  1. 


B.  To  refer  to  any  calendar  day  other  than  the  first  and 
fifteenth  of  each  month,  subtract  the  quantities  below  from  the 
tabular  quantity  for  the  preceding  date. 


Day  of  Month 

Minutes 

No.  of  Days  Elapsed 

2  or  16 

3.9 

1 

3       17 

7.8 

2 

4       18 

11.8 

3 

5       19 

15.7 

4 

6       20 

19.6 

5 

7       21 

23.5 

6 

8       22 

27.4 

7 

9       23 

31.4 

8 

10       24 

35.3 

9 

11       25 

39.2 

10 

12       26 

43.1 

11 

13       27 

47.0 

12 

14       28 

51.0 

13 

29 

54.9 

14 

30 

58.8 

15 

31 

62.7 

16 

Continued  on  page  99 


CULMINATIONS   AND   ELONGATIONS   OF   POLARIS      99 

C.  To  refer  the  table  to  Standard  time  and  to  the  civil  or 
common  method  of  reckoning: 

(a)  Add  to  the  tabular   quantities  four  minutes  for  every 
degree  of  longitude  the  place  is  west  of  the  standard  meridian 
and  subtract  when  the  place  is  east  of  the  standard  meridian. 

(b)  The  astronomical  day  begins  twelve  hours  after  the  civil 
day,  i.e.,  begins  at  noon  on  the  civil  day  of    the  same  date,  and 
is  reckoned  from  zero  to  twenty-four  hours.     Consequently,  an 
astronomical  time  less  than  twelve  hours  refers  to  the  same  civil 
day,  whereas  an  astronomical  time  greater  than  twelve  hours 
refers  to  the  morning  of  the  next  civil  day. 

It  will  be  noticed  that  for  the  tabular  year  two  eastern  elonga- 
tions occur  on  January  14  and  two  western  elongations  on 
July  13.  There  are  also  two  upper  culminations  on  April  14 
and  two  lower  culminations  on  October  14.  The  lower  culmina- 
tion either  follows  or  precedes  the  upper  culminating  by  11 h 
58 -.0. 

D.  To  refer  to  any  other  than  the  tabular  latitude  between 
the  limits  of  10°  and  50°  north: 

Add  to  the  time  of  west  elongation  Om.10  for  every  degree 
south  of  40°  and  subtract  from  the  time  of  west  elongation  Om.16 
for  every  degree  north  of  40°.  Reverse  these  operations  for 
correcting  time  of  east  elongation. 

E.  To  refer  to  any  other  than  the  tabular  longitude: 

Add  Om.16  for  each  15°  east  of  the  ninetieth  meridian  and 
subtract  Om.16  for  each  15°  west  of  the  ninetieth  meridian. 

Table  V  and  the  accompanying  supplementary  tables  and 
rules  have  been  kindly  furnished  for  this  book  by  the  Superin- 
tendent of  the  United  States  Coast  and  Geodetic  Survey. 


100 


MEAN   DECLINATIONS    OF   POLARIS 


TABLE   VI 


MEAN  DECLINATIONS  OF  POLARIS 
For  January  1  of  the  Years  from  1915  to  1928 


Year 

Mean 
Decimation 

Year 

Mean 
Declination 

1915   . 

+88°     51  11' 

1922 

+88°     53  27' 

1916  

88       51.42 

1923  ... 

88       53  57 

1917 

88       51  73 

1924 

88       53  88 

1918.  .  , 

88       52.03 

1925  

88       54  19 

1919 

88       52  34 

1926 

88       54  49 

1920  .  . 

88       52  .  65 

1927.  . 

88       54  80 

1921 

88       52  96 

1928 

88       55  10 

The  above  table  is  based  on  data  obtained  from  the  " American 
Ephemeris  and  Nautical  Almanac."  The  apparent  declination 
of  Polaris  for  any  day  in  the  year  may  be  taken  from  the  " Am- 
erican Ephemeris  and  Nautical  Almanac"  or  from  the  "American 
Nautical  Almanac."  For  1916  the  apparent  declination  de- 
creases from  88°  51'  50".80  on  January  1  to  88°  51'  21".57  on 
June  27,  and  then  increases  to  88°  52'  10".50  on  December  31. 

When  a  possible  error  of  about  half  a  minute  in  declination  is 
too  large  to  be  allowed  the  apparent  declination  should  be 
obtained  from  the  Almanac  for  the  given  date. 


AZIMUTH   OF   PGLAfUS   AT   ELONGATION 


101 


TABLE  VII 

AZIMUTH  OF  POLARIS  WHEN  AT  ELONGATION 
For  Any  Year  Between  1915  and  1928 


Latitude 

1915 

1916 

1917 

1918 

1919 

1920 

1921 

10° 

1°  10.0' 
10.2 
10.4 
10.7 
11.0 
11.3 
11.7 
12.0 
12.4 
12.8 
13.3 
13.8 
14.3 
14.8 
15.4 
16.0 
16.6 
17.3 
18.0 
18.8 
19.6 
20.4 
21.2 
22.1 
23.1 
24.1 
25.2 
26.3 
27.4 
28.6 
29.9 
31.3 
32.7 
34.2 
35.8 
37.4 
39.2 
41.0 
43.0 
45.0 
1°  47.2' 

1°  09.6' 
09.9 
10.1 
10.4 
10.7 
11.0 
11.4 
11.7 
12.1 
12.5 
13.0 
13.5 
14.0 
14.5 
15.1 
15.7 
16.3 
17.0 
17.7 
18.4 
19.2 
20.0 
20.9 
21.8 
22.7 
23.7 
24.8 
25.9 
27.0 
28.2 
29.5 
30.9 
32.3 
33.8 
35.3 
37.0 
38.7 
40.6 
42.5 
44.5 
1°  46.7' 

1°  09.3' 
09.6 
09.8 
10.1 
10.4 
10.7 
11.0 
11.4 
11.8 
12.2 
12.7 
13.1 
13.6 
14.2 
14.7 
15.3 
16.3 
16.6 
17.3 
18.1 
18.8 
19.7 
20.5 
21  A 
22.4 
23.3 
24.4 
25.3 
26.6 
27.8 
29.1 
30.4 
31.9 
33.4 
34.9 
36.6 
38.3 
40.1 
42.0 
44.1 
1°  46.2' 

1°  09.0' 
09.2 
09.5 
09.8 
10.0 
10.4 
10.7 
11.1 
11.5 
11.9 
12.3 
12.8 
]3.3 
13.8 
14.4 
15.0 
15.6 
16.3 
17.0 
17.7 
18.5 
19.3 
20.1 
21.0 
22.0 
23.0 
24.0 
25.1 
26.2 
27.5 
28.7 
30.0 
31.5 
32.9 
34.5 
36.1 
37.8 
39.7 
41.6 
43.6 
1°  45.7 

1°  08.7' 
08.9 
09.2 
09.4 
09.7 
10.0 
10.4 
10.8 
11.1 
11.6 
12.0 
12.5 
13.0 
13.5 
14.1 
14.7 
15.3 
15.9 
16.6 
17.4 
18.1 
18.9 
19.8 
20.7 
21.6 
22.6 
23.6 
24.7 
25.9 
27.1 
28.3 
29.6 
31.0 
32.5 
34.1 
35.7 
37.4 
39.2 
41.1 
43.1 
1°  45.3' 

1°  08.4' 
08.6 
08.9 
09.1 
09.4 
09.7 
10.1 
10.4 
10.7 
11.2 
11.7 
12.2 
12.6 
13.2 
13.7 
14.3 
14.9 
15.6 
16.3 
17.0 
17.8 
18.6 
19.4 
20.3 
21.2 
22.2 
23.3 
24.3 
25.5 
26.7 
27.9 
29.1 
30.6 
32.1 
33.6 
35.3 
37.0 
38.8 
40.7 
42.7 
1°  44.8' 

1°  08.1' 
08.3 
08.6 
08.8 
09.1 
09.4 
09.8 
10.1 
10.5 
10.9 
11.4 
11.8 
12.3 
12.8 
13.4 
14.0 
14.7 
15.2 
15.9 
16.6 
17.4 
18.2 
19.1 
19.9 
20.9 
21.8 
22.9 
24.0 
25.1 
26.3 
27.5 
28.8 
30.2 
31.8 
33.2 
34.8 
36.5 
38.3 
40.2 
42.2 
1°  44.3' 

11  

12 

13  

14 

15  

16 

17  

18  

19  

20  

21    

22 

23   .  

24 

25  

26 

27  

28 

29.  . 

30 

31.  . 

32.  .. 

33.  . 

34  

35.  . 

36 

37.  .. 

38 

39.  . 

40 

41 

42  ...  . 

43 

44.  . 

45 

46  

47 

48  

49.  . 

50  

Continued  on  page  102          Corrections  on  page  103 


102 


AZIMUTH    OF    POLARIS    AT   ELONGATION 


TABLE   VII— Continued 

AZIMUTH  OF  POLARIS  WHEN  AT  ELONGATION 
For  Any  Year  Between  1915  and  1928 


Latitude 

1922 

1923 

1924 

1925 

1926 

1927 

1928 

10° 

1°  07  8' 

1°  07.4' 

1°  07.2' 

1°  06.8' 

1°  06  5' 

1°  06  2' 

1°  05  9' 

11  

08.0 

07.7 

07.4 

07.0 

06.7 

06.4 

06  1 

12 

08  2 

07.9 

07.6 

07.3 

07  0 

06  7 

06  4 

13  

08.5 

08.2 

07.8 

07.6 

07.2 

06  9 

06  6 

14 

08  8 

08.5 

08.2 

07  8 

07  5 

07  2 

06  9 

15  

09.1 

08.8 

08.5 

08.1 

07.8 

07  5 

07  2 

16 

09  4 

09  1 

08  8 

08  5 

08  2 

07  8 

07  5 

17  

09.8 

09.5 

09.2 

08.8 

08  5 

08  2 

07  9 

18 

10  2 

09  8 

09  5 

09  2 

08  9 

08  6 

08  2 

19  

10.6 

10.2 

09.9 

09.6 

09  3 

09  0 

08  6 

20 

11  0 

10  7 

10  4 

10  0 

09  7 

09  4 

09  1 

21  

11.5 

11.2 

10.8 

10.5 

10.2 

09  8 

09  5 

22 

12  0 

11  6 

11  3 

11  0 

10  6 

10  3 

10  0 

23  

12.5 

12.2 

11.8 

11.5 

11.2 

10  8 

10  5 

24    .... 

13  0 

12  7 

12  4 

12  0 

11  7 

11  4 

11  0 

25 

13  6 

13  3 

13  0 

12  6 

12  3 

11  9 

11  6 

26 

14  2 

13  9 

13  6 

13  2 

12  9 

12  5 

12  2 

27 

14  9 

14  6 

14  2 

13  9 

13  5 

13  2 

12  8 

28    .  . 

15  6 

15  2 

14  9 

14  6 

14  2 

13  8 

13  5 

29 

16  3 

16  0 

15  6 

15  2 

14  9 

14  6 

14  2 

30    

17  0 

16  7 

16  4 

16  0 

15  6 

15  3 

14  9 

31 

17  9 

17  5 

17  2 

16  8 

16  4 

16-  1 

15  7 

32  

18  7 

18  3 

18.0 

17.6 

17.2 

16  9 

16  5 

33 

19  6 

19  2 

18  8 

18  5 

18  1 

17  8 

17  4 

34  

20  5 

20  1 

19.8 

19.4 

19.0 

18  6 

18  3 

35 

21  5 

21  1 

20  7 

20  4 

20  0 

19  6 

19  2 

36  

22  5 

22  1 

21.7 

21.4 

21.0 

20  6 

20  2 

37 

23  6 

23  2 

22  8 

22  4 

22  0 

21  6 

21  3 

38  : 

24  7 

24.3 

23.9 

23.5 

23.2 

22.8 

22  4 

39 

25  8 

25  5 

35  1 

24  7 

24  3 

23  9 

23  5 

40  

27.1 

26.7 

26.3 

25.9 

25.5 

25.1 

24  7 

41 

28  4 

28  0 

27  6 

27  2 

26  8 

26  4 

26  0 

42  

29.8 

29.4 

29.0 

28.6 

28.2 

27.8 

27.3 

43 

31  2 

30  8 

30  4 

30  0 

29  6 

29  1 

28  7 

44  

32.8 

32  4 

31.9 

31.5 

£1.1 

30.6 

30.2 

45 

34  4 

34  0 

33  5 

33  1 

32  6 

32  2 

31  8 

46 

36  1 

35  6 

35  2 

34  8 

34  3 

33  9 

33  4 

47  

37  9 

37  4 

37  0 

36  5 

36  1 

35  6 

35  2 

48 

39  8 

39  3 

38  8 

38  4 

37  9 

37  4 

37  0 

49  

41  7 

41  3 

40  8 

40  3 

39  9 

39  4 

38  9 

50 

1°  43  8' 

1°  43  4' 

1°  42  9' 

1°  42  4' 

1°  41  9' 

1°  41  4' 

1°  41  1' 

Corrections  on  page  103 


AZIMUTH   OF   POLARIS   AT   ELONGATION 


103 


The  preceding  table  was  computed  with  the  mean  declination 
of  Polaris  for  each  year.  A  more  accurate  result  will  be  had  by 
applying  to  the  tabular  values  the  following  corrections,  which 
depend  on  the  difference  of  the  mean  and  the  apparent  place  of  the 
star.  The  deduced  azimuth  will,  in  general,  be  correct  within  0'.3. 


For  Middle  of 

Correction 

For  Middle  of 

Correction 

January 

-0.5' 

July  

+0.2' 

February 

—0  4 

August 

+0.1 

March  

-0.3 

September  

-0.1 

April 

0  0 

October 

-0.4 

May  

+0.1 

November  

-0.6 

June 

+0  2 

December           

-0.8 

Table  VII  and  the  accompanying  supplementary  table  have 
been  kindly  furnished  for  this  book  by  the  Superintendent  of 
the  United  States  Coast  and  Geodetic  Survey. 


104 


AZIMUTH    OF    POLARIS    NEAR   ELONGATION 


TABLE  VIII 


CORRECTIONS  FOR  OBTAINING  AZIMUTH  OF  POLARIS  WHEN 
NEAR  ELONGATION  FROM  AZIMUTH  AT  ELONGATION 


Interval 
from 
Elongation 
in 
Minutes 

AZIMUTH  AT  ELONGATION 

1°00' 

1°10' 

1°20' 

1°30' 

1°40' 

1°  50' 

2°  00' 

2°  10' 

2°  20' 

0  . 

0.0" 
0.0 
0.1 
0.3 
0.5 
1 
1 
2 
2 
3 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
14 
15 
17 
18 
20 
21 
23 
25 
27 
28 
30 

0.0" 
0.0 
0.2 
0.4 
0.6 
1 
1 
2 
3 
3 
4 
5 
6 
7 
8 
9 
10 
12 
13 
14 
16 
18 
19 
21 
23 
25 
27 
29 
31 
34 
36 

0.0" 
0.0 
0.2 
0.4 
0.7 
1 
2 
2 
3 
4 
5 
6 
7 
8 
9 
10 
12 
13 
15 
17 
18 
20 
22 
24 
26 
29 
31 
33 
36 
38 
41 

0.0" 
0.1 
0.2 
0.5 
0.8 
1 
2 
3 
3 
4 
5 
6 
7 
9 
10 
12 
13 
15 
17 
19 
21 
23 
25 
27 
29 
32 
35 
38 
40 
43 
46 

0.0" 
0.1 
0.2 
0.5 
0.9 
1 
2 
3 
4 
5 
6 
7 
8 
10 
11 
13 
15 
17 
19 
21 
23 
25 
27 
30 
33 
36 
39 
42 
45 
48 
51 

0.0" 
0.1 
0.3 
0.6 
1.0 
2 
2 
3 
4 
5 
6 
8 
9 
11 
12 
14 
16 
18 
20 
23 
25 
28 
30 
33 
36 
39 
42 
46 
49 
53 
57 

0.0" 
0.1 
0.3 
0.6 
1.1 
2 
3 
3 
4 
6 
7 
8 
10 
12 
13 
15 
18 
20 
22 
25 
27 
30 
33 
36 
40 
43 
46 
50 
54 
58 
62 

0.0" 
0.1 
0.3 
0.7 
1.2 
2 
3 
4 
5 
6 
7 
9 
11 
13 
15 
17 
19 
22 
24 
27 
30 
33 
36 
39 
43 
46 
50 
54 
58 
62 
67 

0.0" 
0.1 
0.3 
0.7 
1.3 
2 
3 
4 
5 
7 
8 
10 
12 
14 
16 
18 
21 
23 
26 
29 
32 
35 
39 
42 
46 
50 
54 
58 
63 
67 
72 

1 

2  

3 

4  

5    .... 

6  

7 

8  

9     .... 

10  

11    .... 

12  

13   

14 

15  

16 

17  

18 

19  

20 

21  

22 

23  

24 

25  

26    .... 

27 

28  

29 

30  

GREEK  ALPHABET 


105 


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Pi 

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Epsilon 

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Rho 

Z,  £ 

Zeta 

2,  <r,  s 

Sigma 

H,  r, 

Eta 

T,    T 

Tau 

0,  0 

Theta 

Y,  v 

Upsilon 

I,    t 

Iota 

*,  * 

Phi 

K,    K 

Kappa 

X,  X 

Chi 

A,  X 

Lambda 

*,  * 

Psi 

M,  /x 

Mu 

0,  o> 

Omega 

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INDEX 


Almanac,  The  American  Nautical,  27       Index  error,  42 


Altitude,  defined,  6 
Apparent  motion,  2 

solar  time,  16 
Artificial  horizon,  45 
Astronomical  time,  19 
triangle,  10 

triangle,  solutions  of,  13 
Atlantic  time,  20 
Autumnal  equinox,  3 
Azimuth,  defined,  6 

determination  of,  55 

mark,  56 

Burt  solar  attachment,  84 

Celestial  equator,  3 

sphere,  1 
Central  time,  20 
Circumpolar  stars,  14,  37 
Civil  time,  19 
Co-latitude,  11 
Constellations,  36 
Co-ordinates,  systems  of,  6 

summary  of,  8 

Corrections  to  observed  altitudes,  43 
Culmination,  16 

Declination,  7 

Eastern  time,  20 
Ecliptic,  3 
Elongation,  15 

Ephemeris,  The  American,  and  Nau- 
tical Almanac,  25 
Equation  of  time,  18 
Equator,  celestial,  3 

systems  of  co-ordinates,  6 
Equinoxes,  3 
Errors  in  observations,  41 

Horizon,  3 

artificial,  45 

system  of  co-ordinates,  6 
Hour  angle,  7 

circle,  3 


Interpolation,  27 

Latitude,  defined,  3 

determination  of,  48 
Local  apparent  time,  16 

mean  time,  17 
Longitude,  defined,  4 

determination  of,  72 

Magnitudes,  36 
Mean  sun,  17 

time,  17 
Meridian,  defined,  3 

determination  of,  see  "Azimuth" 
Motion,  apparent,  2 
Mountain  time,  20 

Nadir,  3 

Nautical  Almanac,  25,  27 

Obliquity  of  the  ecliptic,  3 
Observations,  suggestions  for,  43 
Orbit  of  the  earth,  17 

Pacific  time,  20 
Parallactic  angle,  10 
Parallax,  38 

horizontal,  39 
Pointers,  37 
Polar  distance,  11 
Pole  star  (Polaris),  37 
Poles,  celestial,  3 
Primary  circle,  5 
Prime  vertical,  3 

Refraction,  40 

Relation    between    systems    of    co- 
ordinates, 9 
Right  ascension,  7 
Rotation  of  the  earth,  2 

Saegmuller  solar  attachment,  84,  86 
Secondary  circle,  5 
Semi-diameter,  41 


127 


128 


INDEX 


Sextant,  42,  45 
Sidereal  time,  21 
Solar  attachments,  84 

time,  16,  17 

Spherical  co-ordinates,  5 
summary  of,  8 
trigonometry,  77 
Standard  time,  20 
Stars,  naming  of,  36 

circumpolar,  14,  37 

magnitudes  of,  36 
Sun,  apparent  motion  of,  2,  17 

mean,  17 

Time,  apparent  solar,  16 
astronomical,  19 
civil,  19 

conversion  of,  29 
determination  of,  68 
mean  solar,  17 


Time,  sidereal,  21 

standard,  20 

unit  of  measurement  of,  16 
Transit,  16 

engineer's,  use  of,  41,  43 
Trigonometry,  spherical,  77 
Tropical  year,  22 

Vernal  equinox,  3 
Vertical  circle,  3 
line,  3 

Washington,  longitude  of,  27 
Watch  correction,  68 

Year,  tropical,  22 

Yearly  motion  of  the  sun,  2,  16,  17 

Zenith,  3 

distance,  11 


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)  21-100m-12,'46(A2012sl6)4120 


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